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Theorem el2xpss 8047
Description: Version of elrel 5804 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 3977 . . 3 ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴𝑅) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷))
21ancoms 457 . 2 ((𝐴𝑅𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷))
3 el2xptp 8045 . . 3 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
4 rexex 3073 . . . . . . 7 (∃𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
54reximi 3081 . . . . . 6 (∃𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑦𝐶𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
6 rexex 3073 . . . . . 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 3081 . . . 4 (∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑥𝐵𝑦𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
9 rexex 3073 . . . 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 394   = wceq 1533  wex 1773  wcel 2098  wrex 3067  wss 3949  cotp 4640   × cxp 5680
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-ot 4641  df-iun 5002  df-opab 5215  df-xp 5688  df-rel 5689
This theorem is referenced by:  frxp3  8162
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