Theorem brpprod 34166

Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 34165, 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 34136	. . 3 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2	1	breqi 5084	. 2 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ ⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩)
3		opex 5381	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
4		brpprod.3	. . 3 ⊢ 𝑍 ∈ V
5		brpprod.4	. . 3 ⊢ 𝑊 ∈ V
6	3, 4, 5	brtxp 34161	. 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 5776	. . . 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 5627	. . . . . . . 8 ⊢ ⟨𝑋, 𝑌⟩ ∈ (V × V)
11		vex 3434	. . . . . . . . 9 ⊢ 𝑥 ∈ V
12	11	brresi 5897	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩1st 𝑥))
13	10, 12	mpbiran 705	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
14	8, 9	br1steq 33724	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩1st 𝑥 ↔ 𝑥 = 𝑋)
15	13, 14	bitri 274	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ 𝑥 = 𝑋)
16	15	anbi1i 623	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ (𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
17	16	exbii 1853	. . . 4 ⊢ (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
18		breq1 5081	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝐴𝑍 ↔ 𝑋𝐴𝑍))
19	8, 18	ceqsexv 3477	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
20	7, 17, 19	3bitri 296	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ↔ 𝑋𝐴𝑍)
21	3, 5	brco 5776	. . . 4 ⊢ (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊))
22		vex 3434	. . . . . . . . 9 ⊢ 𝑦 ∈ V
23	22	brresi 5897	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩2nd 𝑦))
24	10, 23	mpbiran 705	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
25	8, 9	br2ndeq 33725	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩2nd 𝑦 ↔ 𝑦 = 𝑌)
26	24, 25	bitri 274	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ 𝑦 = 𝑌)
27	26	anbi1i 623	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ (𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
28	27	exbii 1853	. . . 4 ⊢ (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
29		breq1 5081	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦𝐵𝑊 ↔ 𝑌𝐵𝑊))
30	9, 29	ceqsexv 3477	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
31	21, 28, 30	3bitri 296	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ 𝑌𝐵𝑊)
32	20, 31	anbi12i 626	. 2 ⊢ ((⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ∧ ⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊) ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))
33	2, 6, 32	3bitri 296	1 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1541  ∃wex 1785   ∈ wcel 2109  Vcvv 3430  ⟨cop 4572   class class class wbr 5078   × cxp 5586   ↾ cres 5590   ∘ ccom 5592  1^st c1st 7815  2^nd c2nd 7816   ⊗ ctxp 34111  pprodcpprod 34112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fo 6436  df-fv 6438  df-1st 7817  df-2nd 7818  df-txp 34135  df-pprod 34136

This theorem is referenced by:  brpprod3a  34167