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Theorem dmresexg 5716
 Description: The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
dmresexg (𝐵𝑉 → dom (𝐴𝐵) ∈ V)

Proof of Theorem dmresexg
StepHypRef Expression
1 dmres 5714 . 2 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
2 inex1g 5074 . 2 (𝐵𝑉 → (𝐵 ∩ dom 𝐴) ∈ V)
31, 2syl5eqel 2864 1 (𝐵𝑉 → dom (𝐴𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2048  Vcvv 3409   ∩ cin 3824  dom cdm 5400   ↾ cres 5402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-xp 5406  df-dm 5410  df-res 5412 This theorem is referenced by:  resfunexg  6798  resfunexgALT  7455
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