Theorem brpprod3b 36128

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 36124	. . 3 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)
2	1	breqi 5081	. 2 ⊢ (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ 𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩)
3		brpprod3.1	. . . . 5 ⊢ 𝑋 ∈ V
4		opex 5406	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
5	3, 4	brcnv 5827	. . . 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 36127	. . . 4 ⊢ (⟨𝑌, 𝑍⟩pprod(◡𝑅, ◡𝑆)𝑋 ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤))
9	5, 8	bitri 277	. . 3 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤))
10		biid 263	. . . . 5 ⊢ (𝑋 = ⟨𝑧, 𝑤⟩ ↔ 𝑋 = ⟨𝑧, 𝑤⟩)
11		vex 3437	. . . . . 6 ⊢ 𝑧 ∈ V
12	6, 11	brcnv 5827	. . . . 5 ⊢ (𝑌◡𝑅𝑧 ↔ 𝑧𝑅𝑌)
13		vex 3437	. . . . . 6 ⊢ 𝑤 ∈ V
14	7, 13	brcnv 5827	. . . . 5 ⊢ (𝑍◡𝑆𝑤 ↔ 𝑤𝑆𝑍)
15	10, 12, 14	3anbi123i 1162	. . . 4 ⊢ ((𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ (𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
16	15	2exbii 1857	. . 3 ⊢ (∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
17	9, 16	bitri 277	. 2 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
18	2, 17	bitri 277	1 ⊢ (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))

Syntax hints:   ↔ wb 208   ∧ w3a 1093   = wceq 1548  ∃wex 1787   ∈ wcel 2121  Vcvv 3433  ⟨cop 4564   class class class wbr 5075  ^◡ccnv 5620  pprodcpprod 36072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fo 6495  df-fv 6497  df-1st 7935  df-2nd 7936  df-txp 36095  df-pprod 36096

This theorem is referenced by:  brcart  36173