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Theorem resresdm 40319
 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 5246 . . . 4 (𝐹 = (𝐸𝐴) → dom 𝐹 = dom (𝐸𝐴))
32reseq2d 5317 . . 3 (𝐹 = (𝐸𝐴) → (𝐸 ↾ dom 𝐹) = (𝐸 ↾ dom (𝐸𝐴)))
4 resdmres 5543 . . 3 (𝐸 ↾ dom (𝐸𝐴)) = (𝐸𝐴)
53, 4syl6req 2661 . 2 (𝐹 = (𝐸𝐴) → (𝐸𝐴) = (𝐸 ↾ dom 𝐹))
61, 5eqtrd 2644 1 (𝐹 = (𝐸𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  dom cdm 5038   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050 This theorem is referenced by:  uhgrspan1  40527
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