MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fresaunres1 Structured version   Visualization version   GIF version

Theorem fresaunres1 6763
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 4152 . . 3 (𝐹𝐺) = (𝐺𝐹)
21reseq1i 5976 . 2 ((𝐹𝐺) ↾ 𝐴) = ((𝐺𝐹) ↾ 𝐴)
3 incom 4200 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
43reseq2i 5977 . . . . 5 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐵𝐴))
53reseq2i 5977 . . . . 5 (𝐺 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐵𝐴))
64, 5eqeq12i 2748 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)))
7 eqcom 2737 . . . 4 ((𝐹 ↾ (𝐵𝐴)) = (𝐺 ↾ (𝐵𝐴)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
86, 7bitri 274 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) ↔ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴)))
9 fresaunres2 6762 . . . 4 ((𝐺:𝐵𝐶𝐹:𝐴𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
1093com12 1121 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐺 ↾ (𝐵𝐴)) = (𝐹 ↾ (𝐵𝐴))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
118, 10syl3an3b 1403 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐺𝐹) ↾ 𝐴) = 𝐹)
122, 11eqtrid 2782 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1539  cun 3945  cin 3946  cres 5677  wf 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-dm 5685  df-res 5687  df-fun 6544  df-fn 6545  df-f 6546
This theorem is referenced by:  mapunen  9148  hashf1lem1  14419  hashf1lem1OLD  14420  ptuncnv  23531  resf1o  32222  cvmliftlem10  34583  aacllem  47935
  Copyright terms: Public domain W3C validator