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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 3285 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
2 | vex 3481 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | eldm2 5914 | . . . . 5 ⊢ (𝑦 ∈ dom 𝐵 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵) |
4 | 3 | rexbii 3091 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧〈𝑦, 𝑧〉 ∈ 𝐵) |
5 | eliun 4999 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
6 | 5 | exbii 1844 | . . . 4 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) |
7 | 1, 4, 6 | 3bitr4ri 304 | . . 3 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵) |
8 | 2 | eldm2 5914 | . . 3 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
9 | eliun 4999 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵) | |
10 | 7, 8, 9 | 3bitr4i 303 | . 2 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵) |
11 | 10 | eqriv 2731 | 1 ⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∃wex 1775 ∈ wcel 2105 ∃wrex 3067 〈cop 4636 ∪ ciun 4995 dom cdm 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-iun 4997 df-br 5148 df-dm 5698 |
This theorem is referenced by: dprd2d2 20078 dmdju 32663 gsumpart 33042 esum2d 34073 fmla 35365 iunrelexp0 43691 |
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