Theorem fnresin 32645

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 6705	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))
2		resindi 6025	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵))
3		fnresdm 6699	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
4	3	ineq1d 4240	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ∩ (𝐹 ↾ 𝐵)))
5		incom 4230	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹 ↾ 𝐵))
6		resss 6031	. . . . . . 7 ⊢ (𝐹 ↾ 𝐵) ⊆ 𝐹
7		dfss2 3994	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐵) ⊆ 𝐹 ↔ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵))
8	6, 7	mpbi 230	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵)
9	5, 8	eqtr3i 2770	. . . . 5 ⊢ (𝐹 ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵)
10	4, 9	eqtrdi 2796	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵))
11	2, 10	eqtrid 2792	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
12	11	fneq1d 6672	. 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 1537   ∩ cin 3975   ⊆ wss 3976   ↾ cres 5702   Fn wfn 6568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-fun 6575  df-fn 6576

This theorem is referenced by:  fsuppcurry1  32739  fsuppcurry2  32740  signstres  34552