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Theorem dmresexg 4549
Description: The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
dmresexg (B 𝑉 → dom (AB) V)

Proof of Theorem dmresexg
StepHypRef Expression
1 dmres 4547 . 2 dom (AB) = (B ∩ dom A)
2 inex1g 3856 . 2 (B 𝑉 → (B ∩ dom A) V)
31, 2syl5eqel 2097 1 (B 𝑉 → dom (AB) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1366  Vcvv 2526  cin 2884  dom cdm 4260  cres 4262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-xp 4266  df-dm 4270  df-res 4272
This theorem is referenced by:  resfunexg  5295  resfunexgALT  5648
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