Theorem brpprod3b 31164

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:   𝑤,𝑅,𝑧   𝑤,𝑆,𝑧   𝑤,𝑋,𝑧   𝑤,𝑌,𝑧   𝑤,𝑍,𝑧

Proof of Theorem brpprod3b

Step	Hyp	Ref	Expression
1		pprodcnveq 31160	. . 3 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)
2	1	breqi 4589	. 2 ⊢ (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ 𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩)
3		brpprod3.1	. . . . 5 ⊢ 𝑋 ∈ V
4		opex 4859	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
5	3, 4	brcnv 5227	. . . 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 31163	. . . 4 ⊢ (⟨𝑌, 𝑍⟩pprod(◡𝑅, ◡𝑆)𝑋 ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤))
9	5, 8	bitri 263	. . 3 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤))
10		biid 250	. . . . 5 ⊢ (𝑋 = ⟨𝑧, 𝑤⟩ ↔ 𝑋 = ⟨𝑧, 𝑤⟩)
11		vex 3176	. . . . . 6 ⊢ 𝑧 ∈ V
12	6, 11	brcnv 5227	. . . . 5 ⊢ (𝑌◡𝑅𝑧 ↔ 𝑧𝑅𝑌)
13		vex 3176	. . . . . 6 ⊢ 𝑤 ∈ V
14	7, 13	brcnv 5227	. . . . 5 ⊢ (𝑍◡𝑆𝑤 ↔ 𝑤𝑆𝑍)
15	10, 12, 14	3anbi123i 1244	. . . 4 ⊢ ((𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ (𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
16	15	2exbii 1765	. . 3 ⊢ (∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
17	9, 16	bitri 263	. 2 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
18	2, 17	bitri 263	1 ⊢ (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))

Syntax hints:   ↔ wb 195   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  ^◡ccnv 5037  pprodcpprod 31107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-pprod 31131

This theorem is referenced by:  brcart  31209