Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmresexg Structured version   Visualization version   GIF version

Theorem dmresexg 5852
 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 5850 . 2 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
2 inex1g 5193 . 2 (𝐵𝑉 → (𝐵 ∩ dom 𝐴) ∈ V)
31, 2eqeltrid 2856 1 (𝐵𝑉 → dom (𝐴𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3409   ∩ cin 3859  dom cdm 5528   ↾ cres 5530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-dm 5538  df-res 5540 This theorem is referenced by:  resfunexg  6975  resfunexgALT  7659
 Copyright terms: Public domain W3C validator