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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 2886 | . . . 4 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐶 |
3 | iunxpf.2 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
4 | 3 | nfcri 2886 | . . . 4 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐶 |
5 | iunxpf.3 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
6 | 5 | nfcri 2886 | . . . 4 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷 |
7 | iunxpf.4 | . . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷) | |
8 | 7 | eleq2d 2815 | . . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷)) |
9 | 2, 4, 6, 8 | rexxpf 5850 | . . 3 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
10 | eliun 5000 | . . 3 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶) | |
11 | eliun 5000 | . . . 4 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷) | |
12 | eliun 5000 | . . . . 5 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) | |
13 | 12 | rexbii 3091 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
14 | 11, 13 | bitri 275 | . . 3 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
15 | 9, 10, 14 | 3bitr4i 303 | . 2 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ 𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷) |
16 | 15 | eqriv 2725 | 1 ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Ⅎwnfc 2879 ∃wrex 3067 ⟨cop 4635 ∪ ciun 4996 × cxp 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-iun 4998 df-opab 5211 df-xp 5684 df-rel 5685 |
This theorem is referenced by: dfmpo 8107 pzriprnglem11 21417 |
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