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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 2897 | . . . 4 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐶 |
| 3 | iunxpf.2 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
| 4 | 3 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐶 |
| 5 | iunxpf.3 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
| 6 | 5 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷 |
| 7 | iunxpf.4 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → 𝐶 = 𝐷) | |
| 8 | 7 | eleq2d 2827 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷)) |
| 9 | 2, 4, 6, 8 | rexxpf 5858 | . . 3 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) |
| 10 | eliun 4995 | . . 3 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶) | |
| 11 | eliun 4995 | . . . 4 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷) | |
| 12 | eliun 4995 | . . . . 5 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷) | |
| 13 | 12 | rexbii 3094 | . . . 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 2734 | 1 ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∃wrex 3070 〈cop 4632 ∪ ciun 4991 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: dfmpo 8127 pzriprnglem11 21502 |
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