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Mirrors > Home > MPE Home > Th. List > fvun2 | Structured version Visualization version GIF version |
Description: The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.) |
Ref | Expression |
---|---|
fvun2 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 4168 | . . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹) | |
2 | 1 | fveq1i 6908 | . 2 ⊢ ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐺 ∪ 𝐹)‘𝑋) |
3 | incom 4217 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
4 | 3 | eqeq1i 2740 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅) |
5 | 4 | anbi1i 624 | . . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵) ↔ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵)) |
6 | fvun1 7000 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋)) | |
7 | 5, 6 | syl3an3b 1404 | . . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋)) |
8 | 7 | 3com12 1122 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋)) |
9 | 2, 8 | eqtrid 2787 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 Fn wfn 6558 ‘cfv 6563 |
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-10 2139 ax-11 2155 ax-12 2175 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-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 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-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: fvun2d 7003 fveqf1o 7322 frrlem12 8321 ptunhmeo 23832 noextenddif 27728 noextendlt 27729 noextendgt 27730 noetasuplem4 27796 axlowdimlem9 28980 axlowdimlem12 28983 axlowdimlem17 28988 vtxdun 29514 isoun 32717 resf1o 32748 cycpmfvlem 33115 elrspunidl 33436 lbsdiflsp0 33654 sseqfv2 34376 actfunsnrndisj 34599 reprsuc 34609 breprexplema 34624 cvmliftlem4 35273 fullfunfv 35929 finixpnum 37592 poimirlem1 37608 poimirlem2 37609 poimirlem3 37610 poimirlem4 37611 poimirlem6 37613 poimirlem7 37614 poimirlem11 37618 poimirlem12 37619 poimirlem16 37623 poimirlem19 37626 poimirlem20 37627 poimirlem23 37630 poimirlem28 37635 |
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