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Theorem brpprod3b 34859
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 34855 . . 3 pprod(𝑅, 𝑆) = β—‘pprod(◑𝑅, ◑𝑆)
21breqi 5155 . 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 5465 . . . . 5 βŸ¨π‘Œ, π‘βŸ© ∈ V
53, 4brcnv 5883 . . . 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 34858 . . . 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 3479 . . . . . 6 𝑧 ∈ V
126, 11brcnv 5883 . . . . 5 (π‘Œβ—‘π‘…π‘§ ↔ π‘§π‘…π‘Œ)
13 vex 3479 . . . . . 6 𝑀 ∈ V
147, 13brcnv 5883 . . . . 5 (𝑍◑𝑆𝑀 ↔ 𝑀𝑆𝑍)
1510, 12, 143anbi123i 1156 . . . 4 ((𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘Œβ—‘π‘…π‘§ ∧ 𝑍◑𝑆𝑀) ↔ (𝑋 = βŸ¨π‘§, π‘€βŸ© ∧ π‘§π‘…π‘Œ ∧ 𝑀𝑆𝑍))
16152exbii 1852 . . 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 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3475  βŸ¨cop 4635   class class class wbr 5149  β—‘ccnv 5676  pprodcpprod 34803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-txp 34826  df-pprod 34827
This theorem is referenced by:  brcart  34904
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