Theorem fresin 6696

Description: An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Assertion

Ref	Expression
fresin	⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵)

Proof of Theorem fresin

Step	Hyp	Ref	Expression
1		inss1 4165	. . 3 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝐴
2		fssres 6693	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∩ 𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵)
3	1, 2	mpan2 697	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵)
4		resres 5944	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴 ∩ 𝑋))
5		ffn 6655	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
6		fnresdm 6604	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
7	5, 6	syl 17	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝐴) = 𝐹)
8	7	reseq1d 5930	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ 𝑋))
9	4, 8	eqtr3id 2788	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)) = (𝐹 ↾ 𝑋))
10	9	feq1d 6637	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵 ↔ (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵))
11	3, 10	mpbid 233	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵)

Syntax hints:   → wi 4   = wceq 1547   ∩ cin 3882   ⊆ wss 3883   ↾ cres 5620   Fn wfn 6480  ⟶wf 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-fun 6487  df-fn 6488  df-f 6489

This theorem is referenced by:  o1res  15513  limcresi  25870  dvreslem  25894  dvres2lem  25895  noreson  27642  mbfresfi  38033  ofoafg  43799  limcresiooub  46085  limcresioolb  46086  limcleqr  46087  limclner  46094  mbfres2cn  46401  fouriersw  46674  sge0less  46835  sge0ssre  46840  smfres  47233