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Mirrors > Home > MPE Home > Th. List > iunxpf | Structured version Visualization version GIF version |
Description: Indexed union on a Cartesian product 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 2787 | . . . 4 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐶 |
3 | iunxpf.2 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
4 | 3 | nfcri 2787 | . . . 4 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐶 |
5 | iunxpf.3 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
6 | 5 | nfcri 2787 | . . . 4 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷 |
7 | iunxpf.4 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → 𝐶 = 𝐷) | |
8 | 7 | eleq2d 2716 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷)) |
9 | 2, 4, 6, 8 | rexxpf 5302 | . . 3 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
10 | eliun 4556 | . . 3 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶) | |
11 | eliun 4556 | . . . 4 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷) | |
12 | eliun 4556 | . . . . 5 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) | |
13 | 12 | rexbii 3070 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
14 | 11, 13 | bitri 264 | . . 3 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
15 | 9, 10, 14 | 3bitr4i 292 | . 2 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ 𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷) |
16 | 15 | eqriv 2648 | 1 ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Ⅎwnfc 2780 ∃wrex 2942 〈cop 4216 ∪ ciun 4552 × cxp 5141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-iun 4554 df-opab 4746 df-xp 5149 df-rel 5150 |
This theorem is referenced by: dfmpt2 7312 |
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