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Mirrors > Home > ILE Home > Th. List > iunin2 | GIF version |
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3866 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) |
Ref | Expression |
---|---|
iunin2 | ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.42v 2588 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) | |
2 | elin 3259 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
3 | 2 | rexbii 2442 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) |
4 | eliun 3817 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
5 | 4 | anbi2i 452 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
6 | 1, 3, 5 | 3bitr4i 211 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)) |
7 | eliun 3817 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶)) | |
8 | elin 3259 | . . 3 ⊢ (𝑦 ∈ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)) | |
9 | 6, 7, 8 | 3bitr4i 211 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ 𝑦 ∈ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)) |
10 | 9 | eqriv 2136 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ∩ cin 3070 ∪ ciun 3813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-in 3077 df-iun 3815 |
This theorem is referenced by: iunin1 3877 2iunin 3879 resiun1 4838 resiun2 4839 |
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