Theorem fnresin 30496

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 6460	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))
2		resindi 5844	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵))
3		fnresdm 6454	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
4	3	ineq1d 4118	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ∩ (𝐹 ↾ 𝐵)))
5		incom 4108	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹 ↾ 𝐵))
6		resss 5853	. . . . . . 7 ⊢ (𝐹 ↾ 𝐵) ⊆ 𝐹
7		df-ss 3877	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐵) ⊆ 𝐹 ↔ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵))
8	6, 7	mpbi 233	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵)
9	5, 8	eqtr3i 2783	. . . . 5 ⊢ (𝐹 ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵)
10	4, 9	eqtrdi 2809	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ (𝐹 ↾ 𝐵)) = (𝐹 ↾ 𝐵))
11	2, 10	syl5eq 2805	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
12	11	fneq1d 6432	. 2 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵) ↔ (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵)))
13	1, 12	mpbid 235	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1538   ∩ cin 3859   ⊆ wss 3860   ↾ cres 5530   Fn wfn 6335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-fun 6342  df-fn 6343

This theorem is referenced by:  fsuppcurry1  30596  fsuppcurry2  30597  signstres  32085