Theorem fnresin 32604

Description: Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.)

Assertion

Ref	Expression
fnresin	⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵))

Proof of Theorem fnresin

Step	Hyp	Ref	Expression
1		fnresin1 6663	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))
2		resindi 5982	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵))
3		fnresdm 6657	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
4	3	ineq1d 4194	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ∩ (𝐹 ↾ 𝐵)))
5		incom 4184	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹 ↾ 𝐵))
6		resss 5988	. . . . . . 7 ⊢ (𝐹 ↾ 𝐵) ⊆ 𝐹
7		dfss2 3944	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐵) ⊆ 𝐹 ↔ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵))
8	6, 7	mpbi 230	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵)
9	5, 8	eqtr3i 2760	. . . . 5 ⊢ (𝐹 ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵)
10	4, 9	eqtrdi 2786	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵))
11	2, 10	eqtrid 2782	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
12	11	fneq1d 6631	. 2 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵) ↔ (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵)))
13	1, 12	mpbid 232	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3925   ⊆ wss 3926   ↾ cres 5656   Fn wfn 6526

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-res 5666  df-fun 6533  df-fn 6534

This theorem is referenced by:  fsuppcurry1  32702  fsuppcurry2  32703  signstres  34607