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Theorem brpprod3b 33233
Description: Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
Hypotheses
Ref Expression
brpprod3.1 𝑋 ∈ V
brpprod3.2 𝑌 ∈ V
brpprod3.3 𝑍 ∈ V
Assertion
Ref Expression
brpprod3b (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
Distinct variable groups:   𝑧,𝑤,𝑅   𝑤,𝑆,𝑧   𝑤,𝑋,𝑧   𝑤,𝑌,𝑧   𝑤,𝑍,𝑧

Proof of Theorem brpprod3b
StepHypRef Expression
1 pprodcnveq 33229 . . 3 pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
21breqi 5068 . 2 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ 𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩)
3 brpprod3.1 . . . . 5 𝑋 ∈ V
4 opex 5352 . . . . 5 𝑌, 𝑍⟩ ∈ V
53, 4brcnv 5751 . . . 4 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩pprod(𝑅, 𝑆)𝑋)
6 brpprod3.2 . . . . 5 𝑌 ∈ V
7 brpprod3.3 . . . . 5 𝑍 ∈ V
86, 7, 3brpprod3a 33232 . . . 4 (⟨𝑌, 𝑍⟩pprod(𝑅, 𝑆)𝑋 ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤))
95, 8bitri 276 . . 3 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤))
10 biid 262 . . . . 5 (𝑋 = ⟨𝑧, 𝑤⟩ ↔ 𝑋 = ⟨𝑧, 𝑤⟩)
11 vex 3502 . . . . . 6 𝑧 ∈ V
126, 11brcnv 5751 . . . . 5 (𝑌𝑅𝑧𝑧𝑅𝑌)
13 vex 3502 . . . . . 6 𝑤 ∈ V
147, 13brcnv 5751 . . . . 5 (𝑍𝑆𝑤𝑤𝑆𝑍)
1510, 12, 143anbi123i 1149 . . . 4 ((𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤) ↔ (𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
16152exbii 1842 . . 3 (∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤) ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
179, 16bitri 276 . 2 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
182, 17bitri 276 1 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
Colors of variables: wff setvar class
Syntax hints:  wb 207  w3a 1081   = wceq 1530  wex 1773  wcel 2106  Vcvv 3499  cop 4569   class class class wbr 5062  ccnv 5552  pprodcpprod 33177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-1st 7683  df-2nd 7684  df-txp 33200  df-pprod 33201
This theorem is referenced by:  brcart  33278
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