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

Theorem resresdm 6237
Description: A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.)
Assertion
Ref Expression
resresdm (𝐹 = (𝐸𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))

Proof of Theorem resresdm
StepHypRef Expression
1 id 22 . 2 (𝐹 = (𝐸𝐴) → 𝐹 = (𝐸𝐴))
2 dmeq 5906 . . . 4 (𝐹 = (𝐸𝐴) → dom 𝐹 = dom (𝐸𝐴))
32reseq2d 5985 . . 3 (𝐹 = (𝐸𝐴) → (𝐸 ↾ dom 𝐹) = (𝐸 ↾ dom (𝐸𝐴)))
4 resdmres 6236 . . 3 (𝐸 ↾ dom (𝐸𝐴)) = (𝐸𝐴)
53, 4eqtr2di 2785 . 2 (𝐹 = (𝐸𝐴) → (𝐸𝐴) = (𝐸 ↾ dom 𝐹))
61, 5eqtrd 2768 1 (𝐹 = (𝐸𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  dom cdm 5678  cres 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690
This theorem is referenced by:  uhgrspan1  29115
  Copyright terms: Public domain W3C validator