Theorem pprodss4v 34469

Description: The parallel product is a subclass of ((V × V) × (V × V)). (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
pprodss4v	⊢ pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))

Proof of Theorem pprodss4v

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pprod 34440	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2		txprel 34464	. . 3 ⊢ Rel ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
3		txpss3v 34463	. . . . . . 7 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ (V × (V × V))
4	3	sseli 3940	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
5		opelxp2 5675	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) → 𝑦 ∈ (V × V))
6	4, 5	syl 17	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑦 ∈ (V × V))
7		elvv 5706	. . . . . 6 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
8		opeq2 4831	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ⟨𝑧, 𝑤⟩⟩)
9	8	eleq1d 2822	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ↔ ⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))))
10		df-br 5106	. . . . . . . . 9 ⊢ (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ ↔ ⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))))
11		vex 3449	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
12		vex 3449	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
13		vex 3449	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
14	11, 12, 13	brtxp 34465	. . . . . . . . . 10 ⊢ (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ ↔ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ∧ 𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤))
15	11, 12	brco 5826	. . . . . . . . . . . 12 ⊢ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ↔ ∃𝑦(𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧))
16		vex 3449	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
17	16	brresi 5946	. . . . . . . . . . . . . . 15 ⊢ (𝑥(1st ↾ (V × V))𝑦 ↔ (𝑥 ∈ (V × V) ∧ 𝑥1st 𝑦))
18	17	simplbi 498	. . . . . . . . . . . . . 14 ⊢ (𝑥(1st ↾ (V × V))𝑦 → 𝑥 ∈ (V × V))
19	18	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
20	19	exlimiv 1933	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
21	15, 20	sylbi 216	. . . . . . . . . . 11 ⊢ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 → 𝑥 ∈ (V × V))
22	21	adantr 481	. . . . . . . . . 10 ⊢ ((𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ∧ 𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤) → 𝑥 ∈ (V × V))
23	14, 22	sylbi 216	. . . . . . . . 9 ⊢ (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ → 𝑥 ∈ (V × V))
24	10, 23	sylbir 234	. . . . . . . 8 ⊢ (⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
25	9, 24	syl6bi 252	. . . . . . 7 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
26	25	exlimivv 1935	. . . . . 6 ⊢ (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
27	7, 26	sylbi 216	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
28	6, 27	mpcom 38	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
29	28, 6	opelxpd 5671	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)))
30	2, 29	relssi 5743	. 2 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ ((V × V) × (V × V))
31	1, 30	eqsstri 3978	1 ⊢ pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3445   ⊆ wss 3910  ⟨cop 4592   class class class wbr 5105   × cxp 5631   ↾ cres 5635   ∘ ccom 5637  1^st c1st 7919  2^nd c2nd 7920   ⊗ ctxp 34415  pprodcpprod 34416

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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

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-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-1st 7921  df-2nd 7922  df-txp 34439  df-pprod 34440

This theorem is referenced by:  brpprod3a  34471