Theorem fresaunres1 6380

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:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)

Proof of Theorem fresaunres1

Step	Hyp	Ref	Expression
1		uncom 4018	. . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
2	1	reseq1i 5691	. 2 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐴) = ((𝐺 ∪ 𝐹) ↾ 𝐴)
3		incom 4066	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
4	3	reseq2i 5692	. . . . 5 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐵 ∩ 𝐴))
5	3	reseq2i 5692	. . . . 5 ⊢ (𝐺 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐵 ∩ 𝐴))
6	4, 5	eqeq12i 2792	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) ↔ (𝐹 ↾ (𝐵 ∩ 𝐴)) = (𝐺 ↾ (𝐵 ∩ 𝐴)))
7		eqcom 2785	. . . 4 ⊢ ((𝐹 ↾ (𝐵 ∩ 𝐴)) = (𝐺 ↾ (𝐵 ∩ 𝐴)) ↔ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴)))
8	6, 7	bitri 267	. . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) ↔ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴)))
9		fresaunres2 6379	. . . 4 ⊢ ((𝐺:𝐵⟶𝐶 ∧ 𝐹:𝐴⟶𝐶 ∧ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴))) → ((𝐺 ∪ 𝐹) ↾ 𝐴) = 𝐹)
10	9	3com12 1103	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴))) → ((𝐺 ∪ 𝐹) ↾ 𝐴) = 𝐹)
11	8, 10	syl3an3b 1385	. 2 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐺 ∪ 𝐹) ↾ 𝐴) = 𝐹)
12	2, 11	syl5eq 2826	1 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)

Syntax hints:   → wi 4   ∧ w3a 1068   = wceq 1507   ∪ cun 3827   ∩ cin 3828   ↾ cres 5409  ⟶wf 6184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-dm 5417  df-res 5419  df-fun 6190  df-fn 6191  df-f 6192

This theorem is referenced by:  mapunen  8482  hashf1lem1  13626  ptuncnv  22119  resf1o  30225  cvmliftlem10  32132  aacllem  44275