Theorem fresin2 45159

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 6699	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2	1	eqcomd 2736	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
3	2	ineq2d 4185	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ dom 𝐹))
4	3	reseq2d 5952	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ (𝐶 ∩ dom 𝐹)))
5		frel 6695	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
6		resindm 6003	. . 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 2765	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ 𝐶))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3915  dom cdm 5640   ↾ cres 5642  Rel wrel 5645  ⟶wf 6509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-dm 5650  df-res 5652  df-fun 6515  df-fn 6516  df-f 6517

This theorem is referenced by: (None)