Theorem brpprod3a 33737

Description: Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brpprod3.1	⊢ 𝑋 ∈ V
brpprod3.2	⊢ 𝑌 ∈ V
brpprod3.3	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
brpprod3a	⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))

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

Proof of Theorem brpprod3a

Step	Hyp	Ref	Expression
1		pprodss4v 33735	. . . . . . 7 ⊢ pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V))
2	1	brel 5586	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ 𝑍 ∈ (V × V)))
3	2	simprd 499	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → 𝑍 ∈ (V × V))
4		elvv 5595	. . . . 5 ⊢ (𝑍 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
5	3, 4	sylib 221	. . . 4 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → ∃𝑧∃𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
6	5	pm4.71ri 564	. . 3 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ (∃𝑧∃𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
7		19.41vv 1951	. . 3 ⊢ (∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (∃𝑧∃𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
8	6, 7	bitr4i 281	. 2 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
9		breq2 5036	. . . 4 ⊢ (𝑍 = ⟨𝑧, 𝑤⟩ → (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
10	9	pm5.32i 578	. . 3 ⊢ ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
11	10	2exbii 1850	. 2 ⊢ (∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
12		brpprod3.1	. . . . . 6 ⊢ 𝑋 ∈ V
13		brpprod3.2	. . . . . 6 ⊢ 𝑌 ∈ V
14		vex 3413	. . . . . 6 ⊢ 𝑧 ∈ V
15		vex 3413	. . . . . 6 ⊢ 𝑤 ∈ V
16	12, 13, 14, 15	brpprod 33736	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩ ↔ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))
17	16	anbi2i 625	. . . 4 ⊢ ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)))
18		3anass 1092	. . . 4 ⊢ ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)))
19	17, 18	bitr4i 281	. . 3 ⊢ ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))
20	19	2exbii 1850	. 2 ⊢ (∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))
21	8, 11, 20	3bitri 300	1 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3409  ⟨cop 4528   class class class wbr 5032   × cxp 5522  pprodcpprod 33682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fo 6341  df-fv 6343  df-1st 7693  df-2nd 7694  df-txp 33705  df-pprod 33706

This theorem is referenced by:  brpprod3b  33738  brapply  33789  dfrdg4  33802