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Theorem brpprod 32867
Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 32866, 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 32837 . . 3 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
21breqi 4936 . 2 (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ ⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩)
3 opex 5214 . . 3 𝑋, 𝑌⟩ ∈ V
4 brpprod.3 . . 3 𝑍 ∈ V
5 brpprod.4 . . 3 𝑊 ∈ V
63, 4, 5brtxp 32862 . 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 5592 . . . 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 5447 . . . . . . . 8 𝑋, 𝑌⟩ ∈ (V × V)
11 vex 3418 . . . . . . . . 9 𝑥 ∈ V
1211brresi 5705 . . . . . . . 8 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩1st 𝑥))
1310, 12mpbiran 696 . . . . . . 7 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
148, 9br1steq 32534 . . . . . . 7 (⟨𝑋, 𝑌⟩1st 𝑥𝑥 = 𝑋)
1513, 14bitri 267 . . . . . 6 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥 = 𝑋)
1615anbi1i 614 . . . . 5 ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ (𝑥 = 𝑋𝑥𝐴𝑍))
1716exbii 1810 . . . 4 (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍))
18 breq1 4933 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑍𝑋𝐴𝑍))
198, 18ceqsexv 3462 . . . 4 (∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
207, 17, 193bitri 289 . . 3 (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍𝑋𝐴𝑍)
213, 5brco 5592 . . . 4 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊))
22 vex 3418 . . . . . . . . 9 𝑦 ∈ V
2322brresi 5705 . . . . . . . 8 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩2nd 𝑦))
2410, 23mpbiran 696 . . . . . . 7 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
258, 9br2ndeq 32535 . . . . . . 7 (⟨𝑋, 𝑌⟩2nd 𝑦𝑦 = 𝑌)
2624, 25bitri 267 . . . . . 6 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦 = 𝑌)
2726anbi1i 614 . . . . 5 ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ (𝑦 = 𝑌𝑦𝐵𝑊))
2827exbii 1810 . . . 4 (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊))
29 breq1 4933 . . . . 5 (𝑦 = 𝑌 → (𝑦𝐵𝑊𝑌𝐵𝑊))
309, 29ceqsexv 3462 . . . 4 (∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
3121, 28, 303bitri 289 . . 3 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊𝑌𝐵𝑊)
3220, 31anbi12i 617 . 2 ((⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ∧ ⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊) ↔ (𝑋𝐴𝑍𝑌𝐵𝑊))
332, 6, 323bitri 289 1 (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍𝑌𝐵𝑊))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  wex 1742  wcel 2050  Vcvv 3415  cop 4448   class class class wbr 4930   × cxp 5406  cres 5410  ccom 5412  1st c1st 7501  2nd c2nd 7502  ctxp 32812  pprodcpprod 32813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-fo 6196  df-fv 6198  df-1st 7503  df-2nd 7504  df-txp 32836  df-pprod 32837
This theorem is referenced by:  brpprod3a  32868
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