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Theorem dmresexg 5081
Description: The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
dmresexg  |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )

Proof of Theorem dmresexg
StepHypRef Expression
1 dmres 5079 . 2  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
2 inex1g 4259 . 2  |-  ( B  e.  V  ->  ( B  i^i  dom  A )  e.  _V )
31, 2syl5eqel 2450 1  |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1715   _Vcvv 2873    i^i cin 3237   dom cdm 4792    |` cres 4794
This theorem is referenced by:  resfunexg  5857  resfunexgALT  5858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-xp 4798  df-dm 4802  df-res 4804
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