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Theorem dmresexg 4812
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 4810 . 2  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
2 inex1g 4034 . 2  |-  ( B  e.  V  ->  ( B  i^i  dom  A )  e.  _V )
31, 2eqeltrid 2204 1  |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465   _Vcvv 2660    i^i cin 3040   dom cdm 4509    |` cres 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-dm 4519  df-res 4521
This theorem is referenced by:  resfunexg  5609  resfunexgALT  5976
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