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Theorem brpprod 31669
Description: Characterize a quatary relationship over a tail Cartesian product. Together with pprodss4v 31668, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brpprod.1 𝑋 ∈ V
brpprod.2 𝑌 ∈ V
brpprod.3 𝑍 ∈ V
brpprod.4 𝑊 ∈ V
Assertion
Ref Expression
brpprod (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍𝑌𝐵𝑊))

Proof of Theorem brpprod
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pprod 31638 . . 3 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
21breqi 4624 . 2 (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ ⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩)
3 opex 4898 . . 3 𝑋, 𝑌⟩ ∈ V
4 brpprod.3 . . 3 𝑍 ∈ V
5 brpprod.4 . . 3 𝑊 ∈ V
63, 4, 5brtxp 31664 . 2 (⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩ ↔ (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ∧ ⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊))
73, 4brco 5257 . . . 4 (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ↔ ∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍))
8 brpprod.1 . . . . . . . . 9 𝑋 ∈ V
9 brpprod.2 . . . . . . . . 9 𝑌 ∈ V
108, 9opelvv 5131 . . . . . . . 8 𝑋, 𝑌⟩ ∈ (V × V)
11 vex 3192 . . . . . . . . 9 𝑥 ∈ V
1211brres 5367 . . . . . . . 8 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩1st 𝑥 ∧ ⟨𝑋, 𝑌⟩ ∈ (V × V)))
1310, 12mpbiran2 953 . . . . . . 7 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
148, 9, 11br1steq 31409 . . . . . . 7 (⟨𝑋, 𝑌⟩1st 𝑥𝑥 = 𝑋)
1513, 14bitri 264 . . . . . 6 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥 = 𝑋)
1615anbi1i 730 . . . . 5 ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ (𝑥 = 𝑋𝑥𝐴𝑍))
1716exbii 1771 . . . 4 (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍))
18 breq1 4621 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑍𝑋𝐴𝑍))
198, 18ceqsexv 3231 . . . 4 (∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
207, 17, 193bitri 286 . . 3 (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍𝑋𝐴𝑍)
213, 5brco 5257 . . . 4 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊))
22 vex 3192 . . . . . . . . 9 𝑦 ∈ V
2322brres 5367 . . . . . . . 8 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩2nd 𝑦 ∧ ⟨𝑋, 𝑌⟩ ∈ (V × V)))
2410, 23mpbiran2 953 . . . . . . 7 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
258, 9, 22br2ndeq 31410 . . . . . . 7 (⟨𝑋, 𝑌⟩2nd 𝑦𝑦 = 𝑌)
2624, 25bitri 264 . . . . . 6 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦 = 𝑌)
2726anbi1i 730 . . . . 5 ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ (𝑦 = 𝑌𝑦𝐵𝑊))
2827exbii 1771 . . . 4 (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊))
29 breq1 4621 . . . . 5 (𝑦 = 𝑌 → (𝑦𝐵𝑊𝑌𝐵𝑊))
309, 29ceqsexv 3231 . . . 4 (∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
3121, 28, 303bitri 286 . . 3 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊𝑌𝐵𝑊)
3220, 31anbi12i 732 . 2 ((⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ∧ ⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊) ↔ (𝑋𝐴𝑍𝑌𝐵𝑊))
332, 6, 323bitri 286 1 (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍𝑌𝐵𝑊))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3189  cop 4159   class class class wbr 4618   × cxp 5077  cres 5081  ccom 5083  1st c1st 7118  2nd c2nd 7119  ctxp 31613  pprodcpprod 31614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31637  df-pprod 31638
This theorem is referenced by:  brpprod3a  31670
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