Theorem elxpxpss 33684

Description: Version of elrel 5708 for triple cross products. (Contributed by Scott Fenton, 21-Aug-2024.)

Assertion

Ref	Expression
elxpxpss	⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → ∃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 elxpxpss

Step	Hyp	Ref	Expression
1		ssel2 3916	. 2 ⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷))
2		elxpxp 33683	. . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
3		rexex 3171	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
4	3	reximi 3178	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑦 ∈ 𝐶 ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
5		rexex 3171	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
6	4, 5	syl 17	. . . . 5 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
7	6	reximi 3178	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑥 ∈ 𝐵 ∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
8		rexex 3171	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
9	7, 8	syl 17	. . 3 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
10	2, 9	sylbi 216	. 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
11	1, 10	syl 17	1 ⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065   ⊆ wss 3887  ⟨cop 4567   × cxp 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-iun 4926  df-opab 5137  df-xp 5595  df-rel 5596

This theorem is referenced by:  frxp3  33797