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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 4080 | . . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹) | |
2 | 1 | fveq1i 6646 | . 2 ⊢ ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐺 ∪ 𝐹)‘𝑋) |
3 | incom 4128 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
4 | 3 | eqeq1i 2803 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅) |
5 | 4 | anbi1i 626 | . . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵) ↔ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵)) |
6 | fvun1 6729 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋)) | |
7 | 5, 6 | syl3an3b 1402 | . . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋)) |
8 | 7 | 3com12 1120 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋)) |
9 | 2, 8 | syl5eq 2845 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 Fn wfn 6319 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: fvun2d 6732 fveqf1o 7037 ptunhmeo 22413 axlowdimlem9 26744 axlowdimlem12 26747 axlowdimlem17 26752 vtxdun 27271 isoun 30461 resf1o 30492 cycpmfvlem 30804 elrspunidl 31014 lbsdiflsp0 31110 sseqfv2 31762 actfunsnrndisj 31986 reprsuc 31996 breprexplema 32011 cvmliftlem4 32648 frrlem12 33247 noextenddif 33288 noextendlt 33289 noextendgt 33290 noetalem3 33332 fullfunfv 33521 finixpnum 35042 poimirlem1 35058 poimirlem2 35059 poimirlem3 35060 poimirlem4 35061 poimirlem6 35063 poimirlem7 35064 poimirlem11 35068 poimirlem12 35069 poimirlem16 35073 poimirlem19 35076 poimirlem20 35077 poimirlem23 35080 poimirlem28 35085 |
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