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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 2882 | . . . 4 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐶 |
3 | iunxpf.2 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
4 | 3 | nfcri 2882 | . . . 4 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐶 |
5 | iunxpf.3 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
6 | 5 | nfcri 2882 | . . . 4 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷 |
7 | iunxpf.4 | . . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷) | |
8 | 7 | eleq2d 2811 | . . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷)) |
9 | 2, 4, 6, 8 | rexxpf 5838 | . . 3 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
10 | eliun 4992 | . . 3 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶) | |
11 | eliun 4992 | . . . 4 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷) | |
12 | eliun 4992 | . . . . 5 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) | |
13 | 12 | rexbii 3086 | . . . 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 2721 | 1 ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2875 ∃wrex 3062 ⟨cop 4627 ∪ ciun 4988 × cxp 5665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-iun 4990 df-opab 5202 df-xp 5673 df-rel 5674 |
This theorem is referenced by: dfmpo 8083 pzriprnglem11 21367 |
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