Theorem brpprod 35925

Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 35924, 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 35895	. . 3 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2	1	breqi 5097	. 2 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ ⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩)
3		opex 5404	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
4		brpprod.3	. . 3 ⊢ 𝑍 ∈ V
5		brpprod.4	. . 3 ⊢ 𝑊 ∈ V
6	3, 4, 5	brtxp 35920	. 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 5810	. . . 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 5656	. . . . . . . 8 ⊢ ⟨𝑋, 𝑌⟩ ∈ (V × V)
11		vex 3440	. . . . . . . . 9 ⊢ 𝑥 ∈ V
12	11	brresi 5937	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩1st 𝑥))
13	10, 12	mpbiran 709	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
14	8, 9	br1steq 35813	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩1st 𝑥 ↔ 𝑥 = 𝑋)
15	13, 14	bitri 275	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ 𝑥 = 𝑋)
16	15	anbi1i 624	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ (𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
17	16	exbii 1849	. . . 4 ⊢ (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
18		breq1 5094	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝐴𝑍 ↔ 𝑋𝐴𝑍))
19	8, 18	ceqsexv 3487	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
20	7, 17, 19	3bitri 297	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ↔ 𝑋𝐴𝑍)
21	3, 5	brco 5810	. . . 4 ⊢ (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊))
22		vex 3440	. . . . . . . . 9 ⊢ 𝑦 ∈ V
23	22	brresi 5937	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩2nd 𝑦))
24	10, 23	mpbiran 709	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
25	8, 9	br2ndeq 35814	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩2nd 𝑦 ↔ 𝑦 = 𝑌)
26	24, 25	bitri 275	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ 𝑦 = 𝑌)
27	26	anbi1i 624	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ (𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
28	27	exbii 1849	. . . 4 ⊢ (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
29		breq1 5094	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦𝐵𝑊 ↔ 𝑌𝐵𝑊))
30	9, 29	ceqsexv 3487	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
31	21, 28, 30	3bitri 297	. . 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 297	1 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436  ⟨cop 4582   class class class wbr 5091   × cxp 5614   ↾ cres 5618   ∘ ccom 5620  1^st c1st 7919  2^nd c2nd 7920   ⊗ ctxp 35870  pprodcpprod 35871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fo 6487  df-fv 6489  df-1st 7921  df-2nd 7922  df-txp 35894  df-pprod 35895

This theorem is referenced by:  brpprod3a  35926