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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 4158 | . . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹) | |
| 2 | 1 | fveq1i 6907 | . 2 ⊢ ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐺 ∪ 𝐹)‘𝑋) |
| 3 | incom 4209 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
| 4 | 3 | eqeq1i 2742 | . . . . 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 1407 | . . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋)) |
| 8 | 7 | 3com12 1124 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋)) |
| 9 | 2, 8 | eqtrid 2789 | 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 1087 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 Fn wfn 6556 ‘cfv 6561 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: fvun2d 7003 fveqf1o 7322 frrlem12 8322 ptunhmeo 23816 noextenddif 27713 noextendlt 27714 noextendgt 27715 noetasuplem4 27781 axlowdimlem9 28965 axlowdimlem12 28968 axlowdimlem17 28973 vtxdun 29499 isoun 32711 resf1o 32741 cycpmfvlem 33132 elrspunidl 33456 lbsdiflsp0 33677 sseqfv2 34396 actfunsnrndisj 34620 reprsuc 34630 breprexplema 34645 cvmliftlem4 35293 fullfunfv 35948 finixpnum 37612 poimirlem1 37628 poimirlem2 37629 poimirlem3 37630 poimirlem4 37631 poimirlem6 37633 poimirlem7 37634 poimirlem11 37638 poimirlem12 37639 poimirlem16 37643 poimirlem19 37646 poimirlem20 37647 poimirlem23 37650 poimirlem28 37655 |
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