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Theorem fresaunres1 5974
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 3718 . . 3 (𝐹𝐺) = (𝐺𝐹)
21reseq1i 5299 . 2 ((𝐹𝐺) ↾ 𝐴) = ((𝐺𝐹) ↾ 𝐴)
3 incom 3766 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
43reseq2i 5300 . . . . 5 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐵𝐴))
53reseq2i 5300 . . . . 5 (𝐺 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐵𝐴))
64, 5eqeq12i 2623 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)))
7 eqcom 2616 . . . 4 ((𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
86, 7bitri 262 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
9 fresaunres2 5973 . . . 4 ((𝐺:𝐵𝐶𝐹:𝐴𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
1093com12 1260 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
118, 10syl3an3b 1355 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
122, 11syl5eq 2655 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  cun 3537  cin 3538  cres 5029  wf 5785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-dm 5037  df-res 5039  df-fun 5791  df-fn 5792  df-f 5793
This theorem is referenced by:  mapunen  7991  hashf1lem1  13050  ptuncnv  21367  resf1o  28686  cvmliftlem10  30323  aacllem  42298
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