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

Theorem dmresexg 5560
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 5558 . 2 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
2 inex1g 4935 . 2 (𝐵𝑉 → (𝐵 ∩ dom 𝐴) ∈ V)
31, 2syl5eqel 2854 1 (𝐵𝑉 → dom (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  Vcvv 3351  cin 3722  dom cdm 5249  cres 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-dm 5259  df-res 5261
This theorem is referenced by:  resfunexg  6621  resfunexgALT  7274
  Copyright terms: Public domain W3C validator