Theorem pprodss4v 34925

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 34896	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2		txprel 34920	. . 3 ⊢ Rel ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
3		txpss3v 34919	. . . . . . 7 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ (V × (V × V))
4	3	sseli 3978	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
5		opelxp2 5719	. . . . . 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 5750	. . . . . 6 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
8		opeq2 4874	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ⟨𝑧, 𝑤⟩⟩)
9	8	eleq1d 2818	. . . . . . . 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 5149	. . . . . . . . 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 3478	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
12		vex 3478	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
13		vex 3478	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
14	11, 12, 13	brtxp 34921	. . . . . . . . . 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 5870	. . . . . . . . . . . 12 ⊢ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ↔ ∃𝑦(𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧))
16		vex 3478	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
17	16	brresi 5990	. . . . . . . . . . . . . . 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 5715	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)))
30	2, 29	relssi 5787	. 2 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ ((V × V) × (V × V))
31	1, 30	eqsstri 4016	1 ⊢ pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3948  ⟨cop 4634   class class class wbr 5148   × cxp 5674   ↾ cres 5678   ∘ ccom 5680  1^st c1st 7975  2^nd c2nd 7976   ⊗ ctxp 34871  pprodcpprod 34872

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 5299  ax-nul 5306  ax-pr 5427  ax-un 7727

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7977  df-2nd 7978  df-txp 34895  df-pprod 34896

This theorem is referenced by:  brpprod3a  34927