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Mirrors > Home > MPE Home > Th. List > iunxun | Structured version Visualization version GIF version |
Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
iunxun | ⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexun 4188 | . . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)) | |
2 | eliun 5001 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
3 | eliun 5001 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
4 | 2, 3 | orbi12i 912 | . . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)) |
5 | 1, 4 | bitr4i 277 | . . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)) |
6 | eliun 5001 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶) | |
7 | elun 4145 | . . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)) | |
8 | 5, 6, 7 | 3bitr4i 302 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶)) |
9 | 8 | eqriv 2722 | 1 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 ∪ cun 3942 ∪ ciun 4997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rex 3060 df-v 3463 df-un 3949 df-iun 4999 |
This theorem is referenced by: iunxdif3 5099 iunxprg 5100 iunsuc 6456 funiunfv 7258 iunfi 9371 kmlem11 10190 ackbij1lem9 10258 fsum2dlem 15760 fsumiun 15811 fprod2dlem 15968 prmreclem4 16907 fiuncmp 23369 ovolfiniun 25491 finiunmbl 25534 volfiniun 25537 voliunlem1 25540 uniioombllem4 25576 iuninc 32450 iunxunsn 32456 iunxunpr 32457 ofpreima2 32553 indval2 33784 esum2dlem 33862 sigaclfu2 33891 fiunelros 33944 measvuni 33984 cvmliftlem10 35055 mrsubvrs 35283 mblfinlem2 37282 dfrcl4 43253 iunrelexp0 43279 comptiunov2i 43283 corclrcl 43284 trclfvdecomr 43305 dfrtrcl4 43315 corcltrcl 43316 cotrclrcl 43319 fiiuncl 44576 iunp1 44577 sge0iunmptlemfi 45944 ovolval4lem1 46180 |
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