Theorem fresaunres1 6707

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 4110	. . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
2	1	reseq1i 5934	. 2 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐴) = ((𝐺 ∪ 𝐹) ↾ 𝐴)
3		incom 4161	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
4	3	reseq2i 5935	. . . . 5 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐵 ∩ 𝐴))
5	3	reseq2i 5935	. . . . 5 ⊢ (𝐺 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐵 ∩ 𝐴))
6	4, 5	eqeq12i 2754	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) ↔ (𝐹 ↾ (𝐵 ∩ 𝐴)) = (𝐺 ↾ (𝐵 ∩ 𝐴)))
7		eqcom 2743	. . . 4 ⊢ ((𝐹 ↾ (𝐵 ∩ 𝐴)) = (𝐺 ↾ (𝐵 ∩ 𝐴)) ↔ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴)))
8	6, 7	bitri 275	. . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) ↔ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴)))
9		fresaunres2 6706	. . . 4 ⊢ ((𝐺:𝐵⟶𝐶 ∧ 𝐹:𝐴⟶𝐶 ∧ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴))) → ((𝐺 ∪ 𝐹) ↾ 𝐴) = 𝐹)
10	9	3com12 1123	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴))) → ((𝐺 ∪ 𝐹) ↾ 𝐴) = 𝐹)
11	8, 10	syl3an3b 1407	. 2 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐺 ∪ 𝐹) ↾ 𝐴) = 𝐹)
12	2, 11	eqtrid 2783	1 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∪ cun 3899   ∩ cin 3900   ↾ cres 5626  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-res 5636  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  mapunen  9076  hashf1lem1  14380  ptuncnv  23753  resf1o  32811  cvmliftlem10  35490  aacllem  50067