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Theorem resresdm 5788
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 5480 . . . 4 (𝐹 = (𝐸𝐴) → dom 𝐹 = dom (𝐸𝐴))
32reseq2d 5552 . . 3 (𝐹 = (𝐸𝐴) → (𝐸 ↾ dom 𝐹) = (𝐸 ↾ dom (𝐸𝐴)))
4 resdmres 5787 . . 3 (𝐸 ↾ dom (𝐸𝐴)) = (𝐸𝐴)
53, 4syl6req 2812 . 2 (𝐹 = (𝐸𝐴) → (𝐸𝐴) = (𝐸 ↾ dom 𝐹))
61, 5eqtrd 2795 1 (𝐹 = (𝐸𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  dom cdm 5267  cres 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279
This theorem is referenced by:  uhgrspan1  26416
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