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Theorem fresaunres1 6529
Description: From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.)
Assertion
Ref Expression
fresaunres1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)

Proof of Theorem fresaunres1
StepHypRef Expression
1 uncom 4083 . . 3 (𝐹𝐺) = (𝐺𝐹)
21reseq1i 5818 . 2 ((𝐹𝐺) ↾ 𝐴) = ((𝐺𝐹) ↾ 𝐴)
3 incom 4131 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
43reseq2i 5819 . . . . 5 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐵𝐴))
53reseq2i 5819 . . . . 5 (𝐺 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐵𝐴))
64, 5eqeq12i 2816 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)))
7 eqcom 2808 . . . 4 ((𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
86, 7bitri 278 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
9 fresaunres2 6528 . . . 4 ((𝐺:𝐵𝐶𝐹:𝐴𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
1093com12 1120 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
118, 10syl3an3b 1402 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
122, 11syl5eq 2848 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  cun 3882  cin 3883  cres 5525  wf 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-dm 5533  df-res 5535  df-fun 6330  df-fn 6331  df-f 6332
This theorem is referenced by:  mapunen  8674  hashf1lem1  13813  ptuncnv  22415  resf1o  30495  cvmliftlem10  32649  aacllem  45316
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