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Theorem brpprod3b 34234
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 π‘Œ ∈ V
brpprod3.3 𝑍 ∈ V
Assertion
Ref Expression
brpprod3b (𝑋pprod(𝑅, 𝑆)βŸ¨π‘Œ, π‘βŸ© ↔ βˆƒπ‘§βˆƒπ‘€(𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘§π‘…π‘Œ ∧ 𝑀𝑆𝑍))
Distinct variable groups:   𝑧,𝑀,𝑅   𝑀,𝑆,𝑧   𝑀,𝑋,𝑧   𝑀,π‘Œ,𝑧   𝑀,𝑍,𝑧

Proof of Theorem brpprod3b
StepHypRef Expression
1 pprodcnveq 34230 . . 3 pprod(𝑅, 𝑆) = β—‘pprod(◑𝑅, ◑𝑆)
21breqi 5087 . 2 (𝑋pprod(𝑅, 𝑆)βŸ¨π‘Œ, π‘βŸ© ↔ 𝑋◑pprod(◑𝑅, ◑𝑆)βŸ¨π‘Œ, π‘βŸ©)
3 brpprod3.1 . . . . 5 𝑋 ∈ V
4 opex 5392 . . . . 5 βŸ¨π‘Œ, π‘βŸ© ∈ V
53, 4brcnv 5804 . . . 4 (𝑋◑pprod(◑𝑅, ◑𝑆)βŸ¨π‘Œ, π‘βŸ© ↔ βŸ¨π‘Œ, π‘βŸ©pprod(◑𝑅, ◑𝑆)𝑋)
6 brpprod3.2 . . . . 5 π‘Œ ∈ V
7 brpprod3.3 . . . . 5 𝑍 ∈ V
86, 7, 3brpprod3a 34233 . . . 4 (βŸ¨π‘Œ, π‘βŸ©pprod(◑𝑅, ◑𝑆)𝑋 ↔ βˆƒπ‘§βˆƒπ‘€(𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘Œβ—‘π‘…π‘§ ∧ 𝑍◑𝑆𝑀))
95, 8bitri 275 . . 3 (𝑋◑pprod(◑𝑅, ◑𝑆)βŸ¨π‘Œ, π‘βŸ© ↔ βˆƒπ‘§βˆƒπ‘€(𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘Œβ—‘π‘…π‘§ ∧ 𝑍◑𝑆𝑀))
10 biid 261 . . . . 5 (𝑋 = βŸ¨π‘§, π‘€βŸ© ↔ 𝑋 = βŸ¨π‘§, π‘€βŸ©)
11 vex 3441 . . . . . 6 𝑧 ∈ V
126, 11brcnv 5804 . . . . 5 (π‘Œβ—‘π‘…π‘§ ↔ π‘§π‘…π‘Œ)
13 vex 3441 . . . . . 6 𝑀 ∈ V
147, 13brcnv 5804 . . . . 5 (𝑍◑𝑆𝑀 ↔ 𝑀𝑆𝑍)
1510, 12, 143anbi123i 1155 . . . 4 ((𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘Œβ—‘π‘…π‘§ ∧ 𝑍◑𝑆𝑀) ↔ (𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘§π‘…π‘Œ ∧ 𝑀𝑆𝑍))
16152exbii 1849 . . 3 (βˆƒπ‘§βˆƒπ‘€(𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘Œβ—‘π‘…π‘§ ∧ 𝑍◑𝑆𝑀) ↔ βˆƒπ‘§βˆƒπ‘€(𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘§π‘…π‘Œ ∧ 𝑀𝑆𝑍))
179, 16bitri 275 . 2 (𝑋◑pprod(◑𝑅, ◑𝑆)βŸ¨π‘Œ, π‘βŸ© ↔ βˆƒπ‘§βˆƒπ‘€(𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘§π‘…π‘Œ ∧ 𝑀𝑆𝑍))
182, 17bitri 275 1 (𝑋pprod(𝑅, 𝑆)βŸ¨π‘Œ, π‘βŸ© ↔ βˆƒπ‘§βˆƒπ‘€(𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘§π‘…π‘Œ ∧ 𝑀𝑆𝑍))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ w3a 1087   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  Vcvv 3437  βŸ¨cop 4571   class class class wbr 5081  β—‘ccnv 5599  pprodcpprod 34178
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fo 6464  df-fv 6466  df-1st 7863  df-2nd 7864  df-txp 34201  df-pprod 34202
This theorem is referenced by:  brcart  34279
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