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Theorem brpprod3b 35888
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 35884 . . 3 pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
21breqi 5149 . 2 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ 𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩)
3 brpprod3.1 . . . . 5 𝑋 ∈ V
4 opex 5469 . . . . 5 𝑌, 𝑍⟩ ∈ V
53, 4brcnv 5893 . . . 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 35887 . . . 4 (⟨𝑌, 𝑍⟩pprod(𝑅, 𝑆)𝑋 ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤))
95, 8bitri 275 . . 3 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤))
10 biid 261 . . . . 5 (𝑋 = ⟨𝑧, 𝑤⟩ ↔ 𝑋 = ⟨𝑧, 𝑤⟩)
11 vex 3484 . . . . . 6 𝑧 ∈ V
126, 11brcnv 5893 . . . . 5 (𝑌𝑅𝑧𝑧𝑅𝑌)
13 vex 3484 . . . . . 6 𝑤 ∈ V
147, 13brcnv 5893 . . . . 5 (𝑍𝑆𝑤𝑤𝑆𝑍)
1510, 12, 143anbi123i 1156 . . . 4 ((𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤) ↔ (𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
16152exbii 1849 . . 3 (∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤) ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
179, 16bitri 275 . 2 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
182, 17bitri 275 1 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
Colors of variables: wff setvar class
Syntax hints:  wb 206  w3a 1087   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143  ccnv 5684  pprodcpprod 35832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-pprod 35856
This theorem is referenced by:  brcart  35933
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