Theorem resresdm 6223

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 5894	. . . 4 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → dom 𝐹 = dom (𝐸 ↾ 𝐴))
3	2	reseq2d 5972	. . 3 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ dom 𝐹) = (𝐸 ↾ dom (𝐸 ↾ 𝐴)))
4		resdmres 6222	. . 3 ⊢ (𝐸 ↾ dom (𝐸 ↾ 𝐴)) = (𝐸 ↾ 𝐴)
5	3, 4	eqtr2di 2781	. 2 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ 𝐴) = (𝐸 ↾ dom 𝐹))
6	1, 5	eqtrd 2764	1 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))

Syntax hints:   → wi 4   = wceq 1533  dom cdm 5667   ↾ cres 5669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679

This theorem is referenced by:  uhgrspan1  29053