Theorem brpprod3b 36451

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 36447	. . 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 5443	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
5	3, 4	brcnv 5866	. . . 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 36450	. . . 4 ⊢ (⟨𝑌, 𝑍⟩pprod(◡𝑅, ◡𝑆)𝑋 ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤))
9	5, 8	bitri 278	. . 3 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤))
10		biid 264	. . . . 5 ⊢ (𝑋 = ⟨𝑧, 𝑤⟩ ↔ 𝑋 = ⟨𝑧, 𝑤⟩)
11		vex 3457	. . . . . 6 ⊢ 𝑧 ∈ V
12	6, 11	brcnv 5866	. . . . 5 ⊢ (𝑌◡𝑅𝑧 ↔ 𝑧𝑅𝑌)
13		vex 3457	. . . . . 6 ⊢ 𝑤 ∈ V
14	7, 13	brcnv 5866	. . . . 5 ⊢ (𝑍◡𝑆𝑤 ↔ 𝑤𝑆𝑍)
15	10, 12, 14	3anbi123i 1173	. . . 4 ⊢ ((𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ (𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
16	15	2exbii 1882	. . 3 ⊢ (∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
17	9, 16	bitri 278	. 2 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
18	2, 17	bitri 278	1 ⊢ (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))

This proof depends on syntax axioms:   ↔ wb 209   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3453  ⟨cop 4593   class class class wbr 5107  ^◡ccnv 5658  pprodcpprod 36395

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7989  df-2nd 7990  df-txp 36418  df-pprod 36419

This theorem is used by:  brcart  36496