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Theorem fresin 6541
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 4204 . . 3 (𝐴𝑋) ⊆ 𝐴
2 fssres 6538 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐴𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
31, 2mpan2 687 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
4 resres 5860 . . . 4 ((𝐹𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴𝑋))
5 ffn 6508 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnresdm 6460 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
75, 6syl 17 . . . . 5 (𝐹:𝐴𝐵 → (𝐹𝐴) = 𝐹)
87reseq1d 5846 . . . 4 (𝐹:𝐴𝐵 → ((𝐹𝐴) ↾ 𝑋) = (𝐹𝑋))
94, 8syl5eqr 2870 . . 3 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)) = (𝐹𝑋))
109feq1d 6493 . 2 (𝐹:𝐴𝐵 → ((𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵 ↔ (𝐹𝑋):(𝐴𝑋)⟶𝐵))
113, 10mpbid 233 1 (𝐹:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  cin 3934  wss 3935  cres 5551   Fn wfn 6344  wf 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-fun 6351  df-fn 6352  df-f 6353
This theorem is referenced by:  o1res  14907  limcresi  24412  dvreslem  24436  dvres2lem  24437  noreson  33065  mbfresfi  34820  limcresiooub  41803  limcresioolb  41804  limcleqr  41805  limclner  41812  mbfres2cn  42123  fouriersw  42397  sge0less  42555  sge0ssre  42560  smfres  42946
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