Theorem brpprod 33346

Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 33345, 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 33316	. . 3 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2	1	breqi 5071	. 2 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ ⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩)
3		opex 5355	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
4		brpprod.3	. . 3 ⊢ 𝑍 ∈ V
5		brpprod.4	. . 3 ⊢ 𝑊 ∈ V
6	3, 4, 5	brtxp 33341	. 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 5740	. . . 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 5593	. . . . . . . 8 ⊢ ⟨𝑋, 𝑌⟩ ∈ (V × V)
11		vex 3497	. . . . . . . . 9 ⊢ 𝑥 ∈ V
12	11	brresi 5861	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩1st 𝑥))
13	10, 12	mpbiran 707	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
14	8, 9	br1steq 33014	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩1st 𝑥 ↔ 𝑥 = 𝑋)
15	13, 14	bitri 277	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ 𝑥 = 𝑋)
16	15	anbi1i 625	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ (𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
17	16	exbii 1844	. . . 4 ⊢ (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
18		breq1 5068	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝐴𝑍 ↔ 𝑋𝐴𝑍))
19	8, 18	ceqsexv 3541	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
20	7, 17, 19	3bitri 299	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ↔ 𝑋𝐴𝑍)
21	3, 5	brco 5740	. . . 4 ⊢ (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊))
22		vex 3497	. . . . . . . . 9 ⊢ 𝑦 ∈ V
23	22	brresi 5861	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩2nd 𝑦))
24	10, 23	mpbiran 707	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
25	8, 9	br2ndeq 33015	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩2nd 𝑦 ↔ 𝑦 = 𝑌)
26	24, 25	bitri 277	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ 𝑦 = 𝑌)
27	26	anbi1i 625	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ (𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
28	27	exbii 1844	. . . 4 ⊢ (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
29		breq1 5068	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦𝐵𝑊 ↔ 𝑌𝐵𝑊))
30	9, 29	ceqsexv 3541	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
31	21, 28, 30	3bitri 299	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ 𝑌𝐵𝑊)
32	20, 31	anbi12i 628	. 2 ⊢ ((⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ∧ ⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊) ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))
33	2, 6, 32	3bitri 299	1 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3494  ⟨cop 4572   class class class wbr 5065   × cxp 5552   ↾ cres 5556   ∘ ccom 5558  1^st c1st 7686  2^nd c2nd 7687   ⊗ ctxp 33291  pprodcpprod 33292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-1st 7688  df-2nd 7689  df-txp 33315  df-pprod 33316

This theorem is referenced by:  brpprod3a  33347