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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E24">Th25</font></span>: <a NAME="T25"><span class="comment"><font color="firebrick">:: MESFUN16:25</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font>, <font color="Olive" title="b3">Z</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b4">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b5">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b2">Y</font><br/>  for <font color="Olive" title="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b7">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Olive" title="b3">Z</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b7">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b4">A</font>,<font color="Olive" title="b5">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> holds <br/>( ( <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">A</font> implies  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfun12.html#K2" title="MESFUN12:func.2">ProjPMap1</a> (<font color="Olive" title="b7">f</font>,<font color="Olive" title="b6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">B</font> ) &amp; (  not <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">A</font> implies  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfun12.html#K2" title="MESFUN12:func.2">ProjPMap1</a> (<font color="Olive" title="b7">f</font>,<font color="Olive" title="b6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )</div></div>
