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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 4093 | . . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)) | |
2 | eliun 4835 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
3 | eliun 4835 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
4 | 2, 3 | orbi12i 909 | . . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)) |
5 | 1, 4 | bitr4i 279 | . . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)) |
6 | eliun 4835 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶) | |
7 | elun 4052 | . . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)) | |
8 | 5, 6, 7 | 3bitr4i 304 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶)) |
9 | 8 | eqriv 2794 | 1 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 842 = wceq 1525 ∈ wcel 2083 ∃wrex 3108 ∪ cun 3863 ∪ ciun 4831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-v 3442 df-un 3870 df-iun 4833 |
This theorem is referenced by: iunxdif3 4922 iunxprg 4923 iunsuc 6155 funiunfv 6879 iunfi 8665 kmlem11 9439 ackbij1lem9 9503 fsum2dlem 14962 fsumiun 15013 fprod2dlem 15171 prmreclem4 16088 fiuncmp 21700 ovolfiniun 23789 finiunmbl 23832 volfiniun 23835 voliunlem1 23838 uniioombllem4 23874 iuninc 29998 iunxunsn 30004 iunxunpr 30005 ofpreima2 30097 indval2 30886 esum2dlem 30964 sigaclfu2 30993 fiunelros 31046 measvuni 31086 cvmliftlem10 32151 mrsubvrs 32379 mblfinlem2 34482 dfrcl4 39527 iunrelexp0 39553 comptiunov2i 39557 corclrcl 39558 trclfvdecomr 39579 dfrtrcl4 39589 corcltrcl 39590 cotrclrcl 39593 fiiuncl 40887 iunp1 40888 sge0iunmptlemfi 42259 ovolval4lem1 42495 |
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