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Mirrors > Home > MPE Home > Th. List > dmuni | Structured version Visualization version GIF version |
Description: The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.) |
Ref | Expression |
---|---|
dmuni | ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 2163 | . . . . 5 ⊢ (∃𝑧∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
2 | ancom 462 | . . . . . . 7 ⊢ ((∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥)) | |
3 | 19.41v 1954 | . . . . . . 7 ⊢ (∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
4 | vex 3448 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
5 | 4 | eldm2 5856 | . . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑥 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥) |
6 | 5 | anbi2i 624 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥)) |
7 | 2, 3, 6 | 3bitr4i 303 | . . . . . 6 ⊢ (∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥)) |
8 | 7 | exbii 1851 | . . . . 5 ⊢ (∃𝑥∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥)) |
9 | 1, 8 | bitri 275 | . . . 4 ⊢ (∃𝑧∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥)) |
10 | eluni 4867 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝐴 ↔ ∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
11 | 10 | exbii 1851 | . . . 4 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝐴 ↔ ∃𝑧∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) |
12 | df-rex 3073 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥)) | |
13 | 9, 11, 12 | 3bitr4i 303 | . . 3 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥) |
14 | 4 | eldm2 5856 | . . 3 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝐴) |
15 | eliun 4957 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥) | |
16 | 13, 14, 15 | 3bitr4i 303 | . 2 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥) |
17 | 16 | eqriv 2735 | 1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃wrex 3072 〈cop 4591 ∪ cuni 4864 ∪ ciun 4953 dom cdm 5632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-dm 5642 |
This theorem is referenced by: frrlem7 8216 wfrdmssOLD 8254 wfrdmclOLD 8256 tfrlem8 8323 axdc3lem2 10346 bnj1400 33251 |
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