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Theorem fresin 6778
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 4245 . . 3 (𝐴𝑋) ⊆ 𝐴
2 fssres 6775 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐴𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
31, 2mpan2 691 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
4 resres 6013 . . . 4 ((𝐹𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴𝑋))
5 ffn 6737 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnresdm 6688 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
75, 6syl 17 . . . . 5 (𝐹:𝐴𝐵 → (𝐹𝐴) = 𝐹)
87reseq1d 5999 . . . 4 (𝐹:𝐴𝐵 → ((𝐹𝐴) ↾ 𝑋) = (𝐹𝑋))
94, 8eqtr3id 2789 . . 3 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)) = (𝐹𝑋))
109feq1d 6721 . 2 (𝐹:𝐴𝐵 → ((𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵 ↔ (𝐹𝑋):(𝐴𝑋)⟶𝐵))
113, 10mpbid 232 1 (𝐹:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3962  wss 3963  cres 5691   Fn wfn 6558  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  o1res  15593  limcresi  25935  dvreslem  25959  dvres2lem  25960  noreson  27720  mbfresfi  37653  ofoafg  43344  limcresiooub  45598  limcresioolb  45599  limcleqr  45600  limclner  45607  mbfres2cn  45914  fouriersw  46187  sge0less  46348  sge0ssre  46353  smfres  46746
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