Theorem brpprod 5839

Description: Binary relationship over a parallel product. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
brpprod	⊢ (A PProd (R, S)B ↔ ∃x∃y∃z∃w(A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)))

Distinct variable groups:   w,A,x,y,z   w,B,x,y,z   w,R,x,y,z   w,S,x,y,z

Proof of Theorem brpprod

Step	Hyp	Ref	Expression
1		df-pprod 5738	. . 3 ⊢ PProd (R, S) = ((R ∘ 1st ) ⊗ (S ∘ 2nd ))
2	1	breqi 4645	. 2 ⊢ (A PProd (R, S)B ↔ A((R ∘ 1st ) ⊗ (S ∘ 2nd ))B)
3		brtxp 5783	. 2 ⊢ (A((R ∘ 1st ) ⊗ (S ∘ 2nd ))B ↔ ∃z∃w(B = ⟨z, w⟩ ∧ A(R ∘ 1st )z ∧ A(S ∘ 2nd )w))
4		brco 4883	. . . . . . . 8 ⊢ (A(R ∘ 1st )z ↔ ∃x(A1st x ∧ xRz))
5	4	anbi1i 676	. . . . . . 7 ⊢ ((A(R ∘ 1st )z ∧ A(S ∘ 2nd )w) ↔ (∃x(A1st x ∧ xRz) ∧ A(S ∘ 2nd )w))
6		19.41v 1901	. . . . . . 7 ⊢ (∃x((A1st x ∧ xRz) ∧ A(S ∘ 2nd )w) ↔ (∃x(A1st x ∧ xRz) ∧ A(S ∘ 2nd )w))
7		an32 773	. . . . . . . . 9 ⊢ (((A1st x ∧ xRz) ∧ A(S ∘ 2nd )w) ↔ ((A1st x ∧ A(S ∘ 2nd )w) ∧ xRz))
8		vex 2862	. . . . . . . . . . . . 13 ⊢ x ∈ V
9	8	br1st 4858	. . . . . . . . . . . 12 ⊢ (A1st x ↔ ∃y A = ⟨x, y⟩)
10	9	anbi1i 676	. . . . . . . . . . 11 ⊢ ((A1st x ∧ A(S ∘ 2nd )w) ↔ (∃y A = ⟨x, y⟩ ∧ A(S ∘ 2nd )w))
11		19.41v 1901	. . . . . . . . . . 11 ⊢ (∃y(A = ⟨x, y⟩ ∧ A(S ∘ 2nd )w) ↔ (∃y A = ⟨x, y⟩ ∧ A(S ∘ 2nd )w))
12		breq1 4642	. . . . . . . . . . . . . 14 ⊢ (A = ⟨x, y⟩ → (A(S ∘ 2nd )w ↔ ⟨x, y⟩(S ∘ 2nd )w))
13		vex 2862	. . . . . . . . . . . . . . 15 ⊢ y ∈ V
14	8, 13	brco2nd 5778	. . . . . . . . . . . . . 14 ⊢ (⟨x, y⟩(S ∘ 2nd )w ↔ ySw)
15	12, 14	syl6bb 252	. . . . . . . . . . . . 13 ⊢ (A = ⟨x, y⟩ → (A(S ∘ 2nd )w ↔ ySw))
16	15	pm5.32i 618	. . . . . . . . . . . 12 ⊢ ((A = ⟨x, y⟩ ∧ A(S ∘ 2nd )w) ↔ (A = ⟨x, y⟩ ∧ ySw))
17	16	exbii 1582	. . . . . . . . . . 11 ⊢ (∃y(A = ⟨x, y⟩ ∧ A(S ∘ 2nd )w) ↔ ∃y(A = ⟨x, y⟩ ∧ ySw))
18	10, 11, 17	3bitr2i 264	. . . . . . . . . 10 ⊢ ((A1st x ∧ A(S ∘ 2nd )w) ↔ ∃y(A = ⟨x, y⟩ ∧ ySw))
19	18	anbi1i 676	. . . . . . . . 9 ⊢ (((A1st x ∧ A(S ∘ 2nd )w) ∧ xRz) ↔ (∃y(A = ⟨x, y⟩ ∧ ySw) ∧ xRz))
20		anass 630	. . . . . . . . . . . 12 ⊢ (((A = ⟨x, y⟩ ∧ xRz) ∧ ySw) ↔ (A = ⟨x, y⟩ ∧ (xRz ∧ ySw)))
21		an32 773	. . . . . . . . . . . 12 ⊢ (((A = ⟨x, y⟩ ∧ xRz) ∧ ySw) ↔ ((A = ⟨x, y⟩ ∧ ySw) ∧ xRz))
22	20, 21	bitr3i 242	. . . . . . . . . . 11 ⊢ ((A = ⟨x, y⟩ ∧ (xRz ∧ ySw)) ↔ ((A = ⟨x, y⟩ ∧ ySw) ∧ xRz))
23	22	exbii 1582	. . . . . . . . . 10 ⊢ (∃y(A = ⟨x, y⟩ ∧ (xRz ∧ ySw)) ↔ ∃y((A = ⟨x, y⟩ ∧ ySw) ∧ xRz))
24		19.41v 1901	. . . . . . . . . 10 ⊢ (∃y((A = ⟨x, y⟩ ∧ ySw) ∧ xRz) ↔ (∃y(A = ⟨x, y⟩ ∧ ySw) ∧ xRz))
25	23, 24	bitr2i 241	. . . . . . . . 9 ⊢ ((∃y(A = ⟨x, y⟩ ∧ ySw) ∧ xRz) ↔ ∃y(A = ⟨x, y⟩ ∧ (xRz ∧ ySw)))
26	7, 19, 25	3bitri 262	. . . . . . . 8 ⊢ (((A1st x ∧ xRz) ∧ A(S ∘ 2nd )w) ↔ ∃y(A = ⟨x, y⟩ ∧ (xRz ∧ ySw)))
27	26	exbii 1582	. . . . . . 7 ⊢ (∃x((A1st x ∧ xRz) ∧ A(S ∘ 2nd )w) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (xRz ∧ ySw)))
28	5, 6, 27	3bitr2i 264	. . . . . 6 ⊢ ((A(R ∘ 1st )z ∧ A(S ∘ 2nd )w) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (xRz ∧ ySw)))
29	28	anbi2i 675	. . . . 5 ⊢ ((B = ⟨z, w⟩ ∧ (A(R ∘ 1st )z ∧ A(S ∘ 2nd )w)) ↔ (B = ⟨z, w⟩ ∧ ∃x∃y(A = ⟨x, y⟩ ∧ (xRz ∧ ySw))))
30		3anass 938	. . . . 5 ⊢ ((B = ⟨z, w⟩ ∧ A(R ∘ 1st )z ∧ A(S ∘ 2nd )w) ↔ (B = ⟨z, w⟩ ∧ (A(R ∘ 1st )z ∧ A(S ∘ 2nd )w)))
31		3ancoma 941	. . . . . . . 8 ⊢ ((A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)) ↔ (B = ⟨z, w⟩ ∧ A = ⟨x, y⟩ ∧ (xRz ∧ ySw)))
32		3anass 938	. . . . . . . 8 ⊢ ((B = ⟨z, w⟩ ∧ A = ⟨x, y⟩ ∧ (xRz ∧ ySw)) ↔ (B = ⟨z, w⟩ ∧ (A = ⟨x, y⟩ ∧ (xRz ∧ ySw))))
33	31, 32	bitri 240	. . . . . . 7 ⊢ ((A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)) ↔ (B = ⟨z, w⟩ ∧ (A = ⟨x, y⟩ ∧ (xRz ∧ ySw))))
34	33	2exbii 1583	. . . . . 6 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)) ↔ ∃x∃y(B = ⟨z, w⟩ ∧ (A = ⟨x, y⟩ ∧ (xRz ∧ ySw))))
35		19.42vv 1907	. . . . . 6 ⊢ (∃x∃y(B = ⟨z, w⟩ ∧ (A = ⟨x, y⟩ ∧ (xRz ∧ ySw))) ↔ (B = ⟨z, w⟩ ∧ ∃x∃y(A = ⟨x, y⟩ ∧ (xRz ∧ ySw))))
36	34, 35	bitri 240	. . . . 5 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)) ↔ (B = ⟨z, w⟩ ∧ ∃x∃y(A = ⟨x, y⟩ ∧ (xRz ∧ ySw))))
37	29, 30, 36	3bitr4i 268	. . . 4 ⊢ ((B = ⟨z, w⟩ ∧ A(R ∘ 1st )z ∧ A(S ∘ 2nd )w) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)))
38	37	2exbii 1583	. . 3 ⊢ (∃z∃w(B = ⟨z, w⟩ ∧ A(R ∘ 1st )z ∧ A(S ∘ 2nd )w) ↔ ∃z∃w∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)))
39		exrot4 1745	. . 3 ⊢ (∃z∃w∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)) ↔ ∃x∃y∃z∃w(A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)))
40	38, 39	bitri 240	. 2 ⊢ (∃z∃w(B = ⟨z, w⟩ ∧ A(R ∘ 1st )z ∧ A(S ∘ 2nd )w) ↔ ∃x∃y∃z∃w(A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)))
41	2, 3, 40	3bitri 262	1 ⊢ (A PProd (R, S)B ↔ ∃x∃y∃z∃w(A = ⟨x, y⟩ ∧ B = ⟨z, w⟩ ∧ (xRz ∧ ySw)))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642  ⟨cop 4561   class class class wbr 4639  1^st c1st 4717   ∘ ccom 4721  2^nd c2nd 4783   ⊗ ctxp 5735   PProd cpprod 5737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-co 4726  df-cnv 4785  df-2nd 4797  df-txp 5736  df-pprod 5738

This theorem is referenced by:  dmpprod  5840  fnpprod  5843  frecxp  6314