Theorem resresdm 5882

Description: A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.)

Assertion

Ref	Expression
resresdm	⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))

Proof of Theorem resresdm

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ 𝐴))
2		dmeq 5571	. . . 4 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → dom 𝐹 = dom (𝐸 ↾ 𝐴))
3	2	reseq2d 5644	. . 3 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ dom 𝐹) = (𝐸 ↾ dom (𝐸 ↾ 𝐴)))
4		resdmres 5881	. . 3 ⊢ (𝐸 ↾ dom (𝐸 ↾ 𝐴)) = (𝐸 ↾ 𝐴)
5	3, 4	syl6req 2831	. 2 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ 𝐴) = (𝐸 ↾ dom 𝐹))
6	1, 5	eqtrd 2814	1 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))

Syntax hints:   → wi 4   = wceq 1601  dom cdm 5357   ↾ cres 5359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-dm 5367  df-rn 5368  df-res 5369

This theorem is referenced by:  uhgrspan1  26667