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Theorem fresin 6761
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 4229 . . 3 (𝐴𝑋) ⊆ 𝐴
2 fssres 6758 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐴𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
31, 2mpan2 690 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
4 resres 5995 . . . 4 ((𝐹𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴𝑋))
5 ffn 6718 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnresdm 6670 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
75, 6syl 17 . . . . 5 (𝐹:𝐴𝐵 → (𝐹𝐴) = 𝐹)
87reseq1d 5981 . . . 4 (𝐹:𝐴𝐵 → ((𝐹𝐴) ↾ 𝑋) = (𝐹𝑋))
94, 8eqtr3id 2787 . . 3 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)) = (𝐹𝑋))
109feq1d 6703 . 2 (𝐹:𝐴𝐵 → ((𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵 ↔ (𝐹𝑋):(𝐴𝑋)⟶𝐵))
113, 10mpbid 231 1 (𝐹:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cin 3948  wss 3949  cres 5679   Fn wfn 6539  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  o1res  15504  limcresi  25402  dvreslem  25426  dvres2lem  25427  noreson  27163  mbfresfi  36534  ofoafg  42104  limcresiooub  44358  limcresioolb  44359  limcleqr  44360  limclner  44367  mbfres2cn  44674  fouriersw  44947  sge0less  45108  sge0ssre  45113  smfres  45506
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