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Theorem brpprod3b 32331
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 32327 . . 3 pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
21breqi 4792 . 2 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ 𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩)
3 brpprod3.1 . . . . 5 𝑋 ∈ V
4 opex 5060 . . . . 5 𝑌, 𝑍⟩ ∈ V
53, 4brcnv 5443 . . . 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 32330 . . . 4 (⟨𝑌, 𝑍⟩pprod(𝑅, 𝑆)𝑋 ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤))
95, 8bitri 264 . . 3 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤))
10 biid 251 . . . . 5 (𝑋 = ⟨𝑧, 𝑤⟩ ↔ 𝑋 = ⟨𝑧, 𝑤⟩)
11 vex 3354 . . . . . 6 𝑧 ∈ V
126, 11brcnv 5443 . . . . 5 (𝑌𝑅𝑧𝑧𝑅𝑌)
13 vex 3354 . . . . . 6 𝑤 ∈ V
147, 13brcnv 5443 . . . . 5 (𝑍𝑆𝑤𝑤𝑆𝑍)
1510, 12, 143anbi123i 1158 . . . 4 ((𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤) ↔ (𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
16152exbii 1925 . . 3 (∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤) ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
179, 16bitri 264 . 2 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
182, 17bitri 264 1 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1071   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  cop 4322   class class class wbr 4786  ccnv 5248  pprodcpprod 32275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7315  df-2nd 7316  df-txp 32298  df-pprod 32299
This theorem is referenced by:  brcart  32376
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