Theorem brpprod 36627

Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 36626, 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 36597	. . 3 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2	1	breqi 5109	. 2 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ ⟨𝑋, 𝑌⟩((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))⟨𝑍, 𝑊⟩)
3		opex 5432	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
4		brpprod.3	. . 3 ⊢ 𝑍 ∈ V
5		brpprod.4	. . 3 ⊢ 𝑊 ∈ V
6	3, 4, 5	brtxp 36622	. 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 5848	. . . 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 5691	. . . . . . . 8 ⊢ ⟨𝑋, 𝑌⟩ ∈ (V × V)
11		vex 3455	. . . . . . . . 9 ⊢ 𝑥 ∈ V
12	11	brresi 5979	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩1st 𝑥))
13	10, 12	mpbiran 722	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝑋, 𝑌⟩1st 𝑥)
14	8, 9	br1steq 36515	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩1st 𝑥 ↔ 𝑥 = 𝑋)
15	13, 14	bitri 278	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ↔ 𝑥 = 𝑋)
16	15	anbi1i 636	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ (𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
17	16	exbii 1881	. . . 4 ⊢ (∃𝑥(⟨𝑋, 𝑌⟩(1st ↾ (V × V))𝑥 ∧ 𝑥𝐴𝑍) ↔ ∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍))
18		breq1 5106	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝐴𝑍 ↔ 𝑋𝐴𝑍))
19	8, 18	ceqsexv 3499	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑋 ∧ 𝑥𝐴𝑍) ↔ 𝑋𝐴𝑍)
20	7, 17, 19	3bitri 300	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ↔ 𝑋𝐴𝑍)
21	3, 5	brco 5848	. . . 4 ⊢ (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ ∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊))
22		vex 3455	. . . . . . . . 9 ⊢ 𝑦 ∈ V
23	22	brresi 5979	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ ⟨𝑋, 𝑌⟩2nd 𝑦))
24	10, 23	mpbiran 722	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝑋, 𝑌⟩2nd 𝑦)
25	8, 9	br2ndeq 36516	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩2nd 𝑦 ↔ 𝑦 = 𝑌)
26	24, 25	bitri 278	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ↔ 𝑦 = 𝑌)
27	26	anbi1i 636	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ (𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
28	27	exbii 1881	. . . 4 ⊢ (∃𝑦(⟨𝑋, 𝑌⟩(2nd ↾ (V × V))𝑦 ∧ 𝑦𝐵𝑊) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊))
29		breq1 5106	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦𝐵𝑊 ↔ 𝑌𝐵𝑊))
30	9, 29	ceqsexv 3499	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑌 ∧ 𝑦𝐵𝑊) ↔ 𝑌𝐵𝑊)
31	21, 28, 30	3bitri 300	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊 ↔ 𝑌𝐵𝑊)
32	20, 31	anbi12i 640	. 2 ⊢ ((⟨𝑋, 𝑌⟩(𝐴 ∘ (1st ↾ (V × V)))𝑍 ∧ ⟨𝑋, 𝑌⟩(𝐵 ∘ (2nd ↾ (V × V)))𝑊) ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))
33	2, 6, 32	3bitri 300	1 ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   × cxp 5649   ↾ cres 5653   ∘ ccom 5655  1^st c1st 7997  2^nd c2nd 7998   ⊗ ctxp 36572  pprodcpprod 36573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7999  df-2nd 8000  df-txp 36596  df-pprod 36597

This theorem is used by:  brpprod3a  36628