Theorem fresin2 46460

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 6749	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2	1	eqcomd 2769	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
3	2	ineq2d 4120	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ dom 𝐹))
4	3	reseq2d 5986	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ (𝐶 ∩ dom 𝐹)))
5		resindm 6037	. 2 ⊢ (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹 ↾ 𝐶)
6	4, 5	eqtrdi 2814	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∩ cin 4092  dom cdm 5658   ↾ cres 5660  ⟶wf 6562

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-dm 5668  df-res 5670  df-fn 6569  df-f 6570

This theorem is used by: (None)