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Theorem brpprod3b 32507
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 32503 . . 3 pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
21breqi 4849 . 2 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ 𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩)
3 brpprod3.1 . . . . 5 𝑋 ∈ V
4 opex 5123 . . . . 5 𝑌, 𝑍⟩ ∈ V
53, 4brcnv 5508 . . . 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 32506 . . . 4 (⟨𝑌, 𝑍⟩pprod(𝑅, 𝑆)𝑋 ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤))
95, 8bitri 267 . . 3 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤))
10 biid 253 . . . . 5 (𝑋 = ⟨𝑧, 𝑤⟩ ↔ 𝑋 = ⟨𝑧, 𝑤⟩)
11 vex 3388 . . . . . 6 𝑧 ∈ V
126, 11brcnv 5508 . . . . 5 (𝑌𝑅𝑧𝑧𝑅𝑌)
13 vex 3388 . . . . . 6 𝑤 ∈ V
147, 13brcnv 5508 . . . . 5 (𝑍𝑆𝑤𝑤𝑆𝑍)
1510, 12, 143anbi123i 1195 . . . 4 ((𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤) ↔ (𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
16152exbii 1945 . . 3 (∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌𝑅𝑧𝑍𝑆𝑤) ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
179, 16bitri 267 . 2 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
182, 17bitri 267 1 (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
Colors of variables: wff setvar class
Syntax hints:  wb 198  w3a 1108   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385  cop 4374   class class class wbr 4843  ccnv 5311  pprodcpprod 32451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7401  df-2nd 7402  df-txp 32474  df-pprod 32475
This theorem is referenced by:  brcart  32552
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