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Mirrors > Home > MPE Home > Th. List > el2xpss | Structured version Visualization version GIF version |
Description: Version of elrel 5811 for triple Cartesian products. (Contributed by Scott Fenton, 1-Feb-2025.) |
Ref | Expression |
---|---|
el2xpss | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 3990 | . . 3 ⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷)) | |
2 | 1 | ancoms 458 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷)) |
3 | el2xptp 8059 | . . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | |
4 | rexex 3074 | . . . . . . 7 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉 → ∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | |
5 | 4 | reximi 3082 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉 → ∃𝑦 ∈ 𝐶 ∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
6 | rexex 3074 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉 → ∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉 → ∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
8 | 7 | reximi 3082 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉 → ∃𝑥 ∈ 𝐵 ∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
9 | rexex 3074 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉 → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉 → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
11 | 3, 10 | sylbi 217 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
12 | 2, 11 | syl 17 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 〈cotp 4639 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 df-iun 4998 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: frxp3 8175 |
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