Theorem resresdm 6234

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 23	. 2 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ 𝐴))
2		dmeq 5893	. . . 4 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → dom 𝐹 = dom (𝐸 ↾ 𝐴))
3	2	reseq2d 5978	. . 3 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ dom 𝐹) = (𝐸 ↾ dom (𝐸 ↾ 𝐴)))
4		resdmres 6233	. . 3 ⊢ (𝐸 ↾ dom (𝐸 ↾ 𝐴)) = (𝐸 ↾ 𝐴)
5	3, 4	eqtr2di 2813	. 2 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ 𝐴) = (𝐸 ↾ dom 𝐹))
6	1, 5	eqtrd 2796	1 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))

Syntax hints:   → wi 4   = wceq 1568  dom cdm 5661   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673

This theorem is referenced by:  uhgrspan1  29619