Theorem el2xpss 8047

Description: Version of elrel 5804 for triple Cartesian products. (Contributed by Scott Fenton, 1-Feb-2025.)

Assertion

Ref	Expression
el2xpss	⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → ∃𝑥∃𝑦∃𝑧 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

Step	Hyp	Ref	Expression
1		ssel2 3977	. . 3 ⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷))
2	1	ancoms 457	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷))
3		el2xptp 8045	. . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
4		rexex 3073	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
5	4	reximi 3081	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑦 ∈ 𝐶 ∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
6		rexex 3073	. . . . . 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 3081	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑥 ∈ 𝐵 ∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
9		rexex 3073	. . . 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 216	. 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⟩)

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