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Mirrors > Home > ILE Home > Th. List > dmuni | 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 1599 | . . . . 5 ⊢ (∃𝑧∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
2 | ancom 262 | . . . . . . 7 ⊢ ((∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥)) | |
3 | 19.41v 1830 | . . . . . . 7 ⊢ (∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
4 | vex 2622 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
5 | 4 | eldm2 4634 | . . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑥 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥) |
6 | 5 | anbi2i 445 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧〈𝑦, 𝑧〉 ∈ 𝑥)) |
7 | 2, 3, 6 | 3bitr4i 210 | . . . . . 6 ⊢ (∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥)) |
8 | 7 | exbii 1541 | . . . . 5 ⊢ (∃𝑥∃𝑧(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥)) |
9 | 1, 8 | bitri 182 | . . . 4 ⊢ (∃𝑧∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥)) |
10 | eluni 3656 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝐴 ↔ ∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
11 | 10 | exbii 1541 | . . . 4 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝐴 ↔ ∃𝑧∃𝑥(〈𝑦, 𝑧〉 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) |
12 | df-rex 2365 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥)) | |
13 | 9, 11, 12 | 3bitr4i 210 | . . 3 ⊢ (∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥) |
14 | 4 | eldm2 4634 | . . 3 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ ∃𝑧〈𝑦, 𝑧〉 ∈ ∪ 𝐴) |
15 | eliun 3734 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥) | |
16 | 13, 14, 15 | 3bitr4i 210 | . 2 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥) |
17 | 16 | eqriv 2085 | 1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1289 ∃wex 1426 ∈ wcel 1438 ∃wrex 2360 〈cop 3449 ∪ cuni 3653 ∪ ciun 3730 dom cdm 4438 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-iun 3732 df-br 3846 df-dm 4448 |
This theorem is referenced by: tfrlem8 6083 tfrlemi14d 6098 tfr1onlemres 6114 tfrcllemres 6127 |
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