Theorem fresin 5420

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 3374	. . 3 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝐴
2		fssres 5417	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∩ 𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵)
3	1, 2	mpan2 425	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵)
4		resres 4944	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴 ∩ 𝑋))
5		ffn 5391	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
6		fnresdm 5351	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
7	5, 6	syl 14	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝐴) = 𝐹)
8	7	reseq1d 4931	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ 𝑋))
9	4, 8	eqtr3id 2236	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)) = (𝐹 ↾ 𝑋))
10	9	feq1d 5378	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵 ↔ (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵))
11	3, 10	mpbid 147	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵)

Syntax hints:   → wi 4   = wceq 1364   ∩ cin 3147   ⊆ wss 3148   ↾ cres 4653   Fn wfn 5237  ⟶wf 5238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2758  df-un 3152  df-in 3154  df-ss 3161  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-br 4026  df-opab 4087  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-fun 5244  df-fn 5245  df-f 5246

This theorem is referenced by:  limcresi  14673