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Theorem fresaunres1 6238
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 3900 . . 3 (𝐹𝐺) = (𝐺𝐹)
21reseq1i 5547 . 2 ((𝐹𝐺) ↾ 𝐴) = ((𝐺𝐹) ↾ 𝐴)
3 incom 3948 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
43reseq2i 5548 . . . . 5 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐵𝐴))
53reseq2i 5548 . . . . 5 (𝐺 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐵𝐴))
64, 5eqeq12i 2774 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)))
7 eqcom 2767 . . . 4 ((𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
86, 7bitri 264 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
9 fresaunres2 6237 . . . 4 ((𝐺:𝐵𝐶𝐹:𝐴𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
1093com12 1118 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
118, 10syl3an3b 1512 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
122, 11syl5eq 2806 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  cun 3713  cin 3714  cres 5268  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-dm 5276  df-res 5278  df-fun 6051  df-fn 6052  df-f 6053
This theorem is referenced by:  mapunen  8296  hashf1lem1  13451  ptuncnv  21832  resf1o  29835  cvmliftlem10  31604  aacllem  43078
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