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Theorem brpprod 32117
Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 32116, 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 32087 . . 3 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
21breqi 4691 . 2 (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ ⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩)
3 opex 4962 . . 3 𝑋, 𝑌⟩ ∈ V
4 brpprod.3 . . 3 𝑍 ∈ V
5 brpprod.4 . . 3 𝑊 ∈ V
63, 4, 5brtxp 32112 . 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 5325 . . . 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 5200 . . . . . . . 8 𝑋, 𝑌⟩ ∈ (V × V)
11 vex 3234 . . . . . . . . 9 𝑥 ∈ V
1211brres 5437 . . . . . . . 8 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩1st 𝑥 ∧ ⟨𝑋, 𝑌⟩ ∈ (V × V)))
1310, 12mpbiran2 974 . . . . . . 7 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
148, 9br1steq 31796 . . . . . . 7 (⟨𝑋, 𝑌⟩1st 𝑥𝑥 = 𝑋)
1513, 14bitri 264 . . . . . 6 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥 = 𝑋)
1615anbi1i 731 . . . . 5 ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ (𝑥 = 𝑋𝑥𝐴𝑍))
1716exbii 1814 . . . 4 (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍))
18 breq1 4688 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑍𝑋𝐴𝑍))
198, 18ceqsexv 3273 . . . 4 (∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
207, 17, 193bitri 286 . . 3 (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍𝑋𝐴𝑍)
213, 5brco 5325 . . . 4 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊))
22 vex 3234 . . . . . . . . 9 𝑦 ∈ V
2322brres 5437 . . . . . . . 8 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩2nd 𝑦 ∧ ⟨𝑋, 𝑌⟩ ∈ (V × V)))
2410, 23mpbiran2 974 . . . . . . 7 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
258, 9br2ndeq 31797 . . . . . . 7 (⟨𝑋, 𝑌⟩2nd 𝑦𝑦 = 𝑌)
2624, 25bitri 264 . . . . . 6 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦 = 𝑌)
2726anbi1i 731 . . . . 5 ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ (𝑦 = 𝑌𝑦𝐵𝑊))
2827exbii 1814 . . . 4 (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊))
29 breq1 4688 . . . . 5 (𝑦 = 𝑌 → (𝑦𝐵𝑊𝑌𝐵𝑊))
309, 29ceqsexv 3273 . . . 4 (∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
3121, 28, 303bitri 286 . . 3 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊𝑌𝐵𝑊)
3220, 31anbi12i 733 . 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 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cop 4216   class class class wbr 4685   × cxp 5141  cres 5145  ccom 5147  1st c1st 7208  2nd c2nd 7209  ctxp 32062  pprodcpprod 32063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211  df-txp 32086  df-pprod 32087
This theorem is referenced by:  brpprod3a  32118
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