Theorem fresin2 45412

Description: Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
fresin2	⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ 𝐶))

Proof of Theorem fresin2

Step	Hyp	Ref	Expression
1		fdm 6671	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2	1	eqcomd 2742	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
3	2	ineq2d 4172	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ dom 𝐹))
4	3	reseq2d 5938	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ (𝐶 ∩ dom 𝐹)))
5		frel 6667	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
6		resindm 5989	. . 3 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹 ↾ 𝐶))
7	5, 6	syl 17	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹 ↾ 𝐶))
8	4, 7	eqtrd 2771	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ 𝐶))

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3900  dom cdm 5624   ↾ cres 5626  Rel wrel 5629  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-res 5636  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by: (None)