Theorem brpprod3b 35875

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⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))

Distinct variable groups:   𝑧,𝑤,𝑅   𝑤,𝑆,𝑧   𝑤,𝑋,𝑧   𝑤,𝑌,𝑧   𝑤,𝑍,𝑧

Proof of Theorem brpprod3b

Step	Hyp	Ref	Expression
1		pprodcnveq 35871	. . 3 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)
2	1	breqi 5113	. 2 ⊢ (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ 𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩)
3		brpprod3.1	. . . . 5 ⊢ 𝑋 ∈ V
4		opex 5424	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
5	3, 4	brcnv 5846	. . . 4 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩pprod(◡𝑅, ◡𝑆)𝑋)
6		brpprod3.2	. . . . 5 ⊢ 𝑌 ∈ V
7		brpprod3.3	. . . . 5 ⊢ 𝑍 ∈ V
8	6, 7, 3	brpprod3a 35874	. . . 4 ⊢ (⟨𝑌, 𝑍⟩pprod(◡𝑅, ◡𝑆)𝑋 ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤))
9	5, 8	bitri 275	. . 3 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤))
10		biid 261	. . . . 5 ⊢ (𝑋 = ⟨𝑧, 𝑤⟩ ↔ 𝑋 = ⟨𝑧, 𝑤⟩)
11		vex 3451	. . . . . 6 ⊢ 𝑧 ∈ V
12	6, 11	brcnv 5846	. . . . 5 ⊢ (𝑌◡𝑅𝑧 ↔ 𝑧𝑅𝑌)
13		vex 3451	. . . . . 6 ⊢ 𝑤 ∈ V
14	7, 13	brcnv 5846	. . . . 5 ⊢ (𝑍◡𝑆𝑤 ↔ 𝑤𝑆𝑍)
15	10, 12, 14	3anbi123i 1155	. . . 4 ⊢ ((𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ (𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
16	15	2exbii 1849	. . 3 ⊢ (∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
17	9, 16	bitri 275	. 2 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
18	2, 17	bitri 275	1 ⊢ (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))

Syntax hints:   ↔ wb 206   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   class class class wbr 5107  ^◡ccnv 5637  pprodcpprod 35819

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-1st 7968  df-2nd 7969  df-txp 35842  df-pprod 35843

This theorem is referenced by:  brcart  35920