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Theorem brpprod 35327
Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 35326, 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 35297 . . 3 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
21breqi 5154 . 2 (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ ⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩)
3 opex 5464 . . 3 𝑋, 𝑌⟩ ∈ V
4 brpprod.3 . . 3 𝑍 ∈ V
5 brpprod.4 . . 3 𝑊 ∈ V
63, 4, 5brtxp 35322 . 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 5870 . . . 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 5716 . . . . . . . 8 𝑋, 𝑌⟩ ∈ (V × V)
11 vex 3477 . . . . . . . . 9 𝑥 ∈ V
1211brresi 5990 . . . . . . . 8 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩1st 𝑥))
1310, 12mpbiran 706 . . . . . . 7 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
148, 9br1steq 35212 . . . . . . 7 (⟨𝑋, 𝑌⟩1st 𝑥𝑥 = 𝑋)
1513, 14bitri 275 . . . . . 6 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥 = 𝑋)
1615anbi1i 623 . . . . 5 ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ (𝑥 = 𝑋𝑥𝐴𝑍))
1716exbii 1849 . . . 4 (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍))
18 breq1 5151 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑍𝑋𝐴𝑍))
198, 18ceqsexv 3525 . . . 4 (∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
207, 17, 193bitri 297 . . 3 (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍𝑋𝐴𝑍)
213, 5brco 5870 . . . 4 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊))
22 vex 3477 . . . . . . . . 9 𝑦 ∈ V
2322brresi 5990 . . . . . . . 8 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩2nd 𝑦))
2410, 23mpbiran 706 . . . . . . 7 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
258, 9br2ndeq 35213 . . . . . . 7 (⟨𝑋, 𝑌⟩2nd 𝑦𝑦 = 𝑌)
2624, 25bitri 275 . . . . . 6 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦 = 𝑌)
2726anbi1i 623 . . . . 5 ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ (𝑦 = 𝑌𝑦𝐵𝑊))
2827exbii 1849 . . . 4 (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊))
29 breq1 5151 . . . . 5 (𝑦 = 𝑌 → (𝑦𝐵𝑊𝑌𝐵𝑊))
309, 29ceqsexv 3525 . . . 4 (∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
3121, 28, 303bitri 297 . . 3 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊𝑌𝐵𝑊)
3220, 31anbi12i 626 . 2 ((⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ∧ ⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊) ↔ (𝑋𝐴𝑍𝑌𝐵𝑊))
332, 6, 323bitri 297 1 (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍𝑌𝐵𝑊))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1540  wex 1780  wcel 2105  Vcvv 3473  cop 4634   class class class wbr 5148   × cxp 5674  cres 5678  ccom 5680  1st c1st 7977  2nd c2nd 7978  ctxp 35272  pprodcpprod 35273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7979  df-2nd 7980  df-txp 35296  df-pprod 35297
This theorem is referenced by:  brpprod3a  35328
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