Theorem fresaunres1 6733

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 4121	. . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
2	1	reseq1i 5946	. 2 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐴) = ((𝐺 ∪ 𝐹) ↾ 𝐴)
3		incom 4172	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
4	3	reseq2i 5947	. . . . 5 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ (𝐵 ∩ 𝐴))
5	3	reseq2i 5947	. . . . 5 ⊢ (𝐺 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐵 ∩ 𝐴))
6	4, 5	eqeq12i 2747	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) ↔ (𝐹 ↾ (𝐵 ∩ 𝐴)) = (𝐺 ↾ (𝐵 ∩ 𝐴)))
7		eqcom 2736	. . . 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 6732	. . . 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 2776	1 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∪ cun 3912   ∩ cin 3913   ↾ cres 5640  ⟶wf 6507

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648  df-res 5650  df-fun 6513  df-fn 6514  df-f 6515

This theorem is referenced by:  mapunen  9110  hashf1lem1  14420  ptuncnv  23694  resf1o  32653  cvmliftlem10  35281  aacllem  49790