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Mirrors > Home > MPE Home > Th. List > fresaunres1 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fresaunres1 | ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 4114 | . . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹) | |
2 | 1 | reseq1i 5934 | . 2 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐴) = ((𝐺 ∪ 𝐹) ↾ 𝐴) |
3 | incom 4162 | . . . . . 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 2755 | . . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) ↔ (𝐹 ↾ (𝐵 ∩ 𝐴)) = (𝐺 ↾ (𝐵 ∩ 𝐴))) |
7 | eqcom 2744 | . . . 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 6715 | . . . 4 ⊢ ((𝐺:𝐵⟶𝐶 ∧ 𝐹:𝐴⟶𝐶 ∧ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴))) → ((𝐺 ∪ 𝐹) ↾ 𝐴) = 𝐹) | |
10 | 9 | 3com12 1124 | . . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐺 ↾ (𝐵 ∩ 𝐴)) = (𝐹 ↾ (𝐵 ∩ 𝐴))) → ((𝐺 ∪ 𝐹) ↾ 𝐴) = 𝐹) |
11 | 8, 10 | syl3an3b 1406 | . 2 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐺 ∪ 𝐹) ↾ 𝐴) = 𝐹) |
12 | 2, 11 | eqtrid 2789 | 1 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∪ cun 3909 ∩ cin 3910 ↾ cres 5636 ⟶wf 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-dm 5644 df-res 5646 df-fun 6499 df-fn 6500 df-f 6501 |
This theorem is referenced by: mapunen 9091 hashf1lem1 14354 hashf1lem1OLD 14355 ptuncnv 23161 resf1o 31650 cvmliftlem10 33891 aacllem 47255 |
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