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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 2744 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
2 | vex 2724 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | eldm2 4796 | . . . . 5 ⊢ (𝑦 ∈ dom 𝐵 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵) |
4 | 3 | rexbii 2471 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵) |
5 | eliun 3864 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
6 | 5 | exbii 1592 | . . . 4 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) |
7 | 1, 4, 6 | 3bitr4ri 212 | . . 3 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵) |
8 | 2 | eldm2 4796 | . . 3 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
9 | eliun 3864 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵) | |
10 | 7, 8, 9 | 3bitr4i 211 | . 2 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵) |
11 | 10 | eqriv 2161 | 1 ⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∃wex 1479 ∈ wcel 2135 ∃wrex 2443 〈cop 3573 ∪ ciun 3860 dom cdm 4598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-iun 3862 df-br 3977 df-dm 4608 |
This theorem is referenced by: ennnfonelemdm 12290 |
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