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Theorem resresdm 6077
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 5759 . . . 4 (𝐹 = (𝐸𝐴) → dom 𝐹 = dom (𝐸𝐴))
32reseq2d 5840 . . 3 (𝐹 = (𝐸𝐴) → (𝐸 ↾ dom 𝐹) = (𝐸 ↾ dom (𝐸𝐴)))
4 resdmres 6076 . . 3 (𝐸 ↾ dom (𝐸𝐴)) = (𝐸𝐴)
53, 4syl6req 2876 . 2 (𝐹 = (𝐸𝐴) → (𝐸𝐴) = (𝐸 ↾ dom 𝐹))
61, 5eqtrd 2859 1 (𝐹 = (𝐸𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  dom cdm 5542  cres 5544
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554
This theorem is referenced by:  uhgrspan1  27096
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