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Mirrors > Home > MPE Home > Th. List > fnunres2 | Structured version Visualization version GIF version |
Description: Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023.) |
Ref | Expression |
---|---|
fnunres2 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 4168 | . . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹) | |
2 | 1 | reseq1i 5996 | . 2 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐵) = ((𝐺 ∪ 𝐹) ↾ 𝐵) |
3 | ineqcom 4218 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅) | |
4 | fnunres1 6681 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺) | |
5 | 3, 4 | syl3an3b 1404 | . . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺) |
6 | 5 | 3com12 1122 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺) |
7 | 2, 6 | eqtrid 2787 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 ↾ cres 5691 Fn wfn 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-dm 5699 df-res 5701 df-fun 6565 df-fn 6566 |
This theorem is referenced by: tocycfvres1 33113 cycpmconjslem2 33158 dvun 42368 evlselvlem 42573 evlselv 42574 |
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