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Theorem el2xpss 8019
Description: Version of elrel 5796 for triple Cartesian products. (Contributed by Scott Fenton, 1-Feb-2025.)
Assertion
Ref Expression
el2xpss ((𝐴𝑅𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → ∃𝑥𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧
Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)

Proof of Theorem el2xpss
StepHypRef Expression
1 ssel2 3976 . . 3 ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴𝑅) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷))
21ancoms 459 . 2 ((𝐴𝑅𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷))
3 el2xptp 8017 . . 3 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
4 rexex 3076 . . . . . . 7 (∃𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
54reximi 3084 . . . . . 6 (∃𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑦𝐶𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
6 rexex 3076 . . . . . 6 (∃𝑦𝐶𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
75, 6syl 17 . . . . 5 (∃𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
87reximi 3084 . . . 4 (∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑥𝐵𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
9 rexex 3076 . . . 4 (∃𝑥𝐵𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑥𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
108, 9syl 17 . . 3 (∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑥𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
113, 10sylbi 216 . 2 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → ∃𝑥𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
122, 11syl 17 1 ((𝐴𝑅𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → ∃𝑥𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  wrex 3070  wss 3947  cotp 4635   × cxp 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-ot 4636  df-iun 4998  df-opab 5210  df-xp 5681  df-rel 5682
This theorem is referenced by:  frxp3  8133
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