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Theorem fresin 5099
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:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)

Proof of Theorem fresin
StepHypRef Expression
1 inss1 3193 . . 3 (𝐴𝑋) ⊆ 𝐴
2 fssres 5097 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐴𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
31, 2mpan2 416 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
4 resres 4652 . . . 4 ((𝐹𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴𝑋))
5 ffn 5077 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnresdm 5039 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
75, 6syl 14 . . . . 5 (𝐹:𝐴𝐵 → (𝐹𝐴) = 𝐹)
87reseq1d 4639 . . . 4 (𝐹:𝐴𝐵 → ((𝐹𝐴) ↾ 𝑋) = (𝐹𝑋))
94, 8syl5eqr 2128 . . 3 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)) = (𝐹𝑋))
109feq1d 5065 . 2 (𝐹:𝐴𝐵 → ((𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵 ↔ (𝐹𝑋):(𝐴𝑋)⟶𝐵))
113, 10mpbid 145 1 (𝐹:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  cin 2973  wss 2974  cres 4373   Fn wfn 4927  wf 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-fun 4934  df-fn 4935  df-f 4936
This theorem is referenced by: (None)
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