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Mirrors > Home > ILE Home > Th. List > iunxpf | GIF version |
Description: Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.) |
Ref | Expression |
---|---|
iunxpf.1 | ⊢ Ⅎ𝑦𝐶 |
iunxpf.2 | ⊢ Ⅎ𝑧𝐶 |
iunxpf.3 | ⊢ Ⅎ𝑥𝐷 |
iunxpf.4 | ⊢ (𝑥 = 〈𝑦, 𝑧〉 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
iunxpf | ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxpf.1 | . . . . 5 ⊢ Ⅎ𝑦𝐶 | |
2 | 1 | nfcri 2275 | . . . 4 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐶 |
3 | iunxpf.2 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
4 | 3 | nfcri 2275 | . . . 4 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐶 |
5 | iunxpf.3 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
6 | 5 | nfcri 2275 | . . . 4 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷 |
7 | iunxpf.4 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → 𝐶 = 𝐷) | |
8 | 7 | eleq2d 2209 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷)) |
9 | 2, 4, 6, 8 | rexxpf 4686 | . . 3 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
10 | eliun 3817 | . . 3 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶) | |
11 | eliun 3817 | . . . 4 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷) | |
12 | eliun 3817 | . . . . 5 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) | |
13 | 12 | rexbii 2442 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
14 | 11, 13 | bitri 183 | . . 3 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
15 | 9, 10, 14 | 3bitr4i 211 | . 2 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ 𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷) |
16 | 15 | eqriv 2136 | 1 ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Ⅎwnfc 2268 ∃wrex 2417 〈cop 3530 ∪ ciun 3813 × cxp 4537 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 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-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-iun 3815 df-opab 3990 df-xp 4545 df-rel 4546 |
This theorem is referenced by: dfmpo 6120 |
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