Theorem brpprod3b 35869

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 35865	. . 3 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)
2	1	breqi 5154	. 2 ⊢ (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ 𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩)
3		brpprod3.1	. . . . 5 ⊢ 𝑋 ∈ V
4		opex 5475	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
5	3, 4	brcnv 5896	. . . 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 35868	. . . 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 3482	. . . . . 6 ⊢ 𝑧 ∈ V
12	6, 11	brcnv 5896	. . . . 5 ⊢ (𝑌◡𝑅𝑧 ↔ 𝑧𝑅𝑌)
13		vex 3482	. . . . . 6 ⊢ 𝑤 ∈ V
14	7, 13	brcnv 5896	. . . . 5 ⊢ (𝑍◡𝑆𝑤 ↔ 𝑤𝑆𝑍)
15	10, 12, 14	3anbi123i 1154	. . . 4 ⊢ ((𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ (𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
16	15	2exbii 1846	. . 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 1537  ∃wex 1776   ∈ wcel 2106  Vcvv 3478  ⟨cop 4637   class class class wbr 5148  ^◡ccnv 5688  pprodcpprod 35813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-pprod 35837

This theorem is referenced by:  brcart  35914