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Mirrors > Home > MPE Home > Th. List > dmiun | Structured version Visualization version 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 3198 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
2 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | eldm2 5244 | . . . . 5 ⊢ (𝑦 ∈ dom 𝐵 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵) |
4 | 3 | rexbii 3023 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵) |
5 | eliun 4460 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
6 | 5 | exbii 1764 | . . . 4 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) |
7 | 1, 4, 6 | 3bitr4ri 292 | . . 3 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵) |
8 | 2 | eldm2 5244 | . . 3 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
9 | eliun 4460 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵) | |
10 | 7, 8, 9 | 3bitr4i 291 | . 2 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵) |
11 | 10 | eqriv 2607 | 1 ⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 〈cop 4131 ∪ ciun 4455 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-iun 4457 df-br 4584 df-dm 5048 |
This theorem is referenced by: dprd2d2 18266 esum2d 29482 iunrelexp0 37013 |
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