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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 2162 | . . . . 5 ⊢ (∃𝑧∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
| 2 | ancom 460 | . . . . . . 7 ⊢ ((∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥)) | |
| 3 | 19.41v 1949 | . . . . . . 7 ⊢ (∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
| 4 | vex 3484 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
| 5 | 4 | eldm2 5912 | . . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑥 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥) |
| 6 | 5 | anbi2i 623 | . . . . . . 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 1848 | . . . . 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 4910 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝐴 ↔ ∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
| 11 | 10 | exbii 1848 | . . . 4 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝐴 ↔ ∃𝑧∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) |
| 12 | df-rex 3071 | . . . 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 5912 | . . 3 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝐴) |
| 15 | eliun 4995 | . . 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 2734 | 1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 〈cop 4632 ∪ cuni 4907 ∪ ciun 4991 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-dm 5695 |
| This theorem is referenced by: frrlem7 8317 wfrdmssOLD 8355 wfrdmclOLD 8357 tfrlem8 8424 axdc3lem2 10491 bnj1400 34849 |
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