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Theorem fresin 6588
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 4143 . . 3 (𝐴𝑋) ⊆ 𝐴
2 fssres 6585 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐴𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
31, 2mpan2 691 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵)
4 resres 5864 . . . 4 ((𝐹𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴𝑋))
5 ffn 6545 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnresdm 6496 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
75, 6syl 17 . . . . 5 (𝐹:𝐴𝐵 → (𝐹𝐴) = 𝐹)
87reseq1d 5850 . . . 4 (𝐹:𝐴𝐵 → ((𝐹𝐴) ↾ 𝑋) = (𝐹𝑋))
94, 8eqtr3id 2792 . . 3 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐴𝑋)) = (𝐹𝑋))
109feq1d 6530 . 2 (𝐹:𝐴𝐵 → ((𝐹 ↾ (𝐴𝑋)):(𝐴𝑋)⟶𝐵 ↔ (𝐹𝑋):(𝐴𝑋)⟶𝐵))
113, 10mpbid 235 1 (𝐹:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  cin 3865  wss 3866  cres 5553   Fn wfn 6375  wf 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-fun 6382  df-fn 6383  df-f 6384
This theorem is referenced by:  o1res  15121  limcresi  24782  dvreslem  24806  dvres2lem  24807  noreson  33600  mbfresfi  35560  limcresiooub  42858  limcresioolb  42859  limcleqr  42860  limclner  42867  mbfres2cn  43174  fouriersw  43447  sge0less  43605  sge0ssre  43610  smfres  43996
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