Theorem pprodss4v 33458

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 33429	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2		txprel 33453	. . 3 ⊢ Rel ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
3		txpss3v 33452	. . . . . . 7 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ (V × (V × V))
4	3	sseli 3911	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
5		opelxp2 5561	. . . . . 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 5590	. . . . . 6 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
8		opeq2 4765	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ⟨𝑧, 𝑤⟩⟩)
9	8	eleq1d 2874	. . . . . . . 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 5031	. . . . . . . . 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 3444	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
12		vex 3444	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
13		vex 3444	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
14	11, 12, 13	brtxp 33454	. . . . . . . . . 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 5705	. . . . . . . . . . . 12 ⊢ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ↔ ∃𝑦(𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧))
16		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
17	16	brresi 5827	. . . . . . . . . . . . . . 15 ⊢ (𝑥(1st ↾ (V × V))𝑦 ↔ (𝑥 ∈ (V × V) ∧ 𝑥1st 𝑦))
18	17	simplbi 501	. . . . . . . . . . . . . 14 ⊢ (𝑥(1st ↾ (V × V))𝑦 → 𝑥 ∈ (V × V))
19	18	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
20	19	exlimiv 1931	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
21	15, 20	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 → 𝑥 ∈ (V × V))
22	21	adantr 484	. . . . . . . . . 10 ⊢ ((𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ∧ 𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤) → 𝑥 ∈ (V × V))
23	14, 22	sylbi 220	. . . . . . . . 9 ⊢ (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ → 𝑥 ∈ (V × V))
24	10, 23	sylbir 238	. . . . . . . 8 ⊢ (⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
25	9, 24	syl6bi 256	. . . . . . 7 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
26	25	exlimivv 1933	. . . . . 6 ⊢ (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
27	7, 26	sylbi 220	. . . . 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 5557	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)))
30	2, 29	relssi 5624	. 2 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ ((V × V) × (V × V))
31	1, 30	eqsstri 3949	1 ⊢ pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  ⟨cop 4531   class class class wbr 5030   × cxp 5517   ↾ cres 5521   ∘ ccom 5523  1^st c1st 7669  2^nd c2nd 7670   ⊗ ctxp 33404  pprodcpprod 33405

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-1st 7671  df-2nd 7672  df-txp 33428  df-pprod 33429

This theorem is referenced by:  brpprod3a  33460