Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brpprod Structured version   Visualization version   GIF version

Theorem brpprod 35867
Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 35866, 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 35837 . . 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 5475 . . 3 𝑋, 𝑌⟩ ∈ V
4 brpprod.3 . . 3 𝑍 ∈ V
5 brpprod.4 . . 3 𝑊 ∈ V
63, 4, 5brtxp 35862 . 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 5884 . . . 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 5729 . . . . . . . 8 𝑋, 𝑌⟩ ∈ (V × V)
11 vex 3482 . . . . . . . . 9 𝑥 ∈ V
1211brresi 6009 . . . . . . . 8 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩1st 𝑥))
1310, 12mpbiran 709 . . . . . . 7 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
148, 9br1steq 35752 . . . . . . 7 (⟨𝑋, 𝑌⟩1st 𝑥𝑥 = 𝑋)
1513, 14bitri 275 . . . . . 6 (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥 = 𝑋)
1615anbi1i 624 . . . . 5 ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ (𝑥 = 𝑋𝑥𝐴𝑍))
1716exbii 1845 . . . 4 (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍))
18 breq1 5151 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑍𝑋𝐴𝑍))
198, 18ceqsexv 3530 . . . 4 (∃𝑥(𝑥 = 𝑋𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
207, 17, 193bitri 297 . . 3 (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍𝑋𝐴𝑍)
213, 5brco 5884 . . . 4 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊))
22 vex 3482 . . . . . . . . 9 𝑦 ∈ V
2322brresi 6009 . . . . . . . 8 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩2nd 𝑦))
2410, 23mpbiran 709 . . . . . . 7 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
258, 9br2ndeq 35753 . . . . . . 7 (⟨𝑋, 𝑌⟩2nd 𝑦𝑦 = 𝑌)
2624, 25bitri 275 . . . . . 6 (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦 = 𝑌)
2726anbi1i 624 . . . . 5 ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ (𝑦 = 𝑌𝑦𝐵𝑊))
2827exbii 1845 . . . 4 (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊))
29 breq1 5151 . . . . 5 (𝑦 = 𝑌 → (𝑦𝐵𝑊𝑌𝐵𝑊))
309, 29ceqsexv 3530 . . . 4 (∃𝑦(𝑦 = 𝑌𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
3121, 28, 303bitri 297 . . 3 (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊𝑌𝐵𝑊)
3220, 31anbi12i 628 . 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 206  wa 395   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148   × cxp 5687  cres 5691  ccom 5693  1st c1st 8011  2nd c2nd 8012  ctxp 35812  pprodcpprod 35813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-pprod 35837
This theorem is referenced by:  brpprod3a  35868
  Copyright terms: Public domain W3C validator