Theorem brpprod 32867

Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 32866, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brpprod.1	⊢ 𝑋 ∈ V
brpprod.2	⊢ 𝑌 ∈ V
brpprod.3	⊢ 𝑍 ∈ V
brpprod.4	⊢ 𝑊 ∈ V

Assertion

Ref	Expression
brpprod	⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))

Proof of Theorem brpprod

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pprod 32837	. . 3 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2	1	breqi 4936	. 2 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ ⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩)
3		opex 5214	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
4		brpprod.3	. . 3 ⊢ 𝑍 ∈ V
5		brpprod.4	. . 3 ⊢ 𝑊 ∈ V
6	3, 4, 5	brtxp 32862	. 2 ⊢ (⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩ ↔ (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ∧ ⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊))
7	3, 4	brco 5592	. . . 4 ⊢ (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ↔ ∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍))
8		brpprod.1	. . . . . . . . 9 ⊢ 𝑋 ∈ V
9		brpprod.2	. . . . . . . . 9 ⊢ 𝑌 ∈ V
10	8, 9	opelvv 5447	. . . . . . . 8 ⊢ ⟨𝑋, 𝑌⟩ ∈ (V × V)
11		vex 3418	. . . . . . . . 9 ⊢ 𝑥 ∈ V
12	11	brresi 5705	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩1st 𝑥))
13	10, 12	mpbiran 696	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
14	8, 9	br1steq 32534	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩1st 𝑥 ↔ 𝑥 = 𝑋)
15	13, 14	bitri 267	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ 𝑥 = 𝑋)
16	15	anbi1i 614	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ (𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
17	16	exbii 1810	. . . 4 ⊢ (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
18		breq1 4933	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝐴𝑍 ↔ 𝑋𝐴𝑍))
19	8, 18	ceqsexv 3462	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
20	7, 17, 19	3bitri 289	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ↔ 𝑋𝐴𝑍)
21	3, 5	brco 5592	. . . 4 ⊢ (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊))
22		vex 3418	. . . . . . . . 9 ⊢ 𝑦 ∈ V
23	22	brresi 5705	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩2nd 𝑦))
24	10, 23	mpbiran 696	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
25	8, 9	br2ndeq 32535	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩2nd 𝑦 ↔ 𝑦 = 𝑌)
26	24, 25	bitri 267	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ 𝑦 = 𝑌)
27	26	anbi1i 614	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ (𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
28	27	exbii 1810	. . . 4 ⊢ (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
29		breq1 4933	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦𝐵𝑊 ↔ 𝑌𝐵𝑊))
30	9, 29	ceqsexv 3462	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
31	21, 28, 30	3bitri 289	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ 𝑌𝐵𝑊)
32	20, 31	anbi12i 617	. 2 ⊢ ((⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ∧ ⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊) ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))
33	2, 6, 32	3bitri 289	1 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))

Syntax hints:   ↔ wb 198   ∧ wa 387   = wceq 1507  ∃wex 1742   ∈ wcel 2050  Vcvv 3415  ⟨cop 4448   class class class wbr 4930   × cxp 5406   ↾ cres 5410   ∘ ccom 5412  1^st c1st 7501  2^nd c2nd 7502   ⊗ ctxp 32812  pprodcpprod 32813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-fo 6196  df-fv 6198  df-1st 7503  df-2nd 7504  df-txp 32836  df-pprod 32837

This theorem is referenced by:  brpprod3a  32868