Theorem pprodss4v 31991

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.)

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 31962	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2		txprel 31986	. . 3 ⊢ Rel ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
3		txpss3v 31985	. . . . . . 7 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ (V × (V × V))
4	3	sseli 3599	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
5		opelxp2 5151	. . . . . 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 5177	. . . . . 6 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
8		opeq2 4403	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ⟨𝑧, 𝑤⟩⟩)
9	8	eleq1d 2686	. . . . . . . 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 4654	. . . . . . . . 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 3203	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
12		vex 3203	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
13		vex 3203	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
14	11, 12, 13	brtxp 31987	. . . . . . . . . 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 5292	. . . . . . . . . . . 12 ⊢ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ↔ ∃𝑦(𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧))
16		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
17	16	brres 5402	. . . . . . . . . . . . . . 15 ⊢ (𝑥(1st ↾ (V × V))𝑦 ↔ (𝑥1st 𝑦 ∧ 𝑥 ∈ (V × V)))
18	17	simprbi 480	. . . . . . . . . . . . . 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 1858	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑥(1st ↾ (V × V))𝑦 ∧ 𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
21	15, 20	sylbi 207	. . . . . . . . . . 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 207	. . . . . . . . 9 ⊢ (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ → 𝑥 ∈ (V × V))
24	10, 23	sylbir 225	. . . . . . . 8 ⊢ (⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
25	9, 24	syl6bi 243	. . . . . . 7 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
26	25	exlimivv 1860	. . . . . 6 ⊢ (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
27	7, 26	sylbi 207	. . . . 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		opelxp 5146	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)) ↔ (𝑥 ∈ (V × V) ∧ 𝑦 ∈ (V × V)))
30	28, 6, 29	sylanbrc 698	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)))
31	2, 30	relssi 5211	. 2 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ ((V × V) × (V × V))
32	1, 31	eqsstri 3635	1 ⊢ pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ⟨cop 4183   class class class wbr 4653   × cxp 5112   ↾ cres 5116   ∘ ccom 5118  1^st c1st 7166  2^nd c2nd 7167   ⊗ ctxp 31937  pprodcpprod 31938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-pprod 31962

This theorem is referenced by:  brpprod3a  31993