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Mirrors > Home > ILE Home > Th. List > dmiun | GIF version |
Description: The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
dmiun | ⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 2622 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
2 | vex 2604 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | eldm2 4551 | . . . . 5 ⊢ (𝑦 ∈ dom 𝐵 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵) |
4 | 3 | rexbii 2373 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵) |
5 | eliun 3682 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
6 | 5 | exbii 1536 | . . . 4 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) |
7 | 1, 4, 6 | 3bitr4ri 211 | . . 3 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵) |
8 | 2 | eldm2 4551 | . . 3 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
9 | eliun 3682 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵) | |
10 | 7, 8, 9 | 3bitr4i 210 | . 2 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵) |
11 | 10 | eqriv 2078 | 1 ⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∃wrex 2349 〈cop 3401 ∪ ciun 3678 dom cdm 4363 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-iun 3680 df-br 3786 df-dm 4373 |
This theorem is referenced by: (None) |
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