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Theorem pprodss4v 33419
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.
StepHypRef Expression
1 df-pprod 33390 . 2 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2 txprel 33414 . . 3 Rel ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
3 txpss3v 33413 . . . . . . 7 ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ (V × (V × V))
43sseli 3938 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
5 opelxp2 5574 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) → 𝑦 ∈ (V × V))
64, 5syl 17 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑦 ∈ (V × V))
7 elvv 5603 . . . . . 6 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
8 opeq2 4778 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ⟨𝑧, 𝑤⟩⟩)
98eleq1d 2898 . . . . . . . 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 5043 . . . . . . . . 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 3472 . . . . . . . . . . 11 𝑥 ∈ V
12 vex 3472 . . . . . . . . . . 11 𝑧 ∈ V
13 vex 3472 . . . . . . . . . . 11 𝑤 ∈ V
1411, 12, 13brtxp 33415 . . . . . . . . . 10 (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ ↔ (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤))
1511, 12brco 5718 . . . . . . . . . . . 12 (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧 ↔ ∃𝑦(𝑥(1st ↾ (V × V))𝑦𝑦𝐴𝑧))
16 vex 3472 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
1716brresi 5840 . . . . . . . . . . . . . . 15 (𝑥(1st ↾ (V × V))𝑦 ↔ (𝑥 ∈ (V × V) ∧ 𝑥1st 𝑦))
1817simplbi 501 . . . . . . . . . . . . . 14 (𝑥(1st ↾ (V × V))𝑦𝑥 ∈ (V × V))
1918adantr 484 . . . . . . . . . . . . 13 ((𝑥(1st ↾ (V × V))𝑦𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
2019exlimiv 1931 . . . . . . . . . . . 12 (∃𝑦(𝑥(1st ↾ (V × V))𝑦𝑦𝐴𝑧) → 𝑥 ∈ (V × V))
2115, 20sylbi 220 . . . . . . . . . . 11 (𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧𝑥 ∈ (V × V))
2221adantr 484 . . . . . . . . . 10 ((𝑥(𝐴 ∘ (1st ↾ (V × V)))𝑧𝑥(𝐵 ∘ (2nd ↾ (V × V)))𝑤) → 𝑥 ∈ (V × V))
2314, 22sylbi 220 . . . . . . . . 9 (𝑥((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑧, 𝑤⟩ → 𝑥 ∈ (V × V))
2410, 23sylbir 238 . . . . . . . 8 (⟨𝑥, ⟨𝑧, 𝑤⟩⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
259, 24syl6bi 256 . . . . . . 7 (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
2625exlimivv 1933 . . . . . 6 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
277, 26sylbi 220 . . . . 5 (𝑦 ∈ (V × V) → (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V)))
286, 27mpcom 38 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → 𝑥 ∈ (V × V))
2928, 6opelxpd 5570 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) → ⟨𝑥, 𝑦⟩ ∈ ((V × V) × (V × V)))
302, 29relssi 5637 . 2 ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) ⊆ ((V × V) × (V × V))
311, 30eqsstri 3976 1 pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2114  Vcvv 3469  wss 3908  cop 4545   class class class wbr 5042   × cxp 5530  cres 5534  ccom 5536  1st c1st 7673  2nd c2nd 7674  ctxp 33365  pprodcpprod 33366
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fo 6340  df-fv 6342  df-1st 7675  df-2nd 7676  df-txp 33389  df-pprod 33390
This theorem is referenced by:  brpprod3a  33421
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