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Theorem fresaunres1 6751
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 4149 . . 3 (𝐹𝐺) = (𝐺𝐹)
21reseq1i 5969 . 2 ((𝐹𝐺) ↾ 𝐴) = ((𝐺𝐹) ↾ 𝐴)
3 incom 4197 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
43reseq2i 5970 . . . . 5 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐵𝐴))
53reseq2i 5970 . . . . 5 (𝐺 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐵𝐴))
64, 5eqeq12i 2749 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)))
7 eqcom 2738 . . . 4 ((𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
86, 7bitri 274 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
9 fresaunres2 6750 . . . 4 ((𝐺:𝐵𝐶𝐹:𝐴𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
1093com12 1123 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
118, 10syl3an3b 1405 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
122, 11eqtrid 2783 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  cun 3942  cin 3943  cres 5671  wf 6528
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-dm 5679  df-res 5681  df-fun 6534  df-fn 6535  df-f 6536
This theorem is referenced by:  mapunen  9129  hashf1lem1  14397  hashf1lem1OLD  14398  ptuncnv  23240  resf1o  31826  cvmliftlem10  34116  aacllem  47496
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