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Theorem dmresexg 5144
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 5142 . 2  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
2 inex1g 4565 . 2  |-  ( B  e.  V  ->  ( B  i^i  dom  A )  e.  _V )
31, 2syl5eqel 2515 1  |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1869   _Vcvv 3082    i^i cin 3436   dom cdm 4851    |` cres 4853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-xp 4857  df-dm 4861  df-res 4863
This theorem is referenced by:  resfunexg  6143  resfunexgALT  6768
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