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Theorem brpprod 5839
 Description: Binary relationship over a parallel product. (Contributed by SF, 24-Feb-2015.)
Assertion
Ref Expression
brpprod (A PProd (R, S)Bxyzw(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
StepHypRef Expression
1 df-pprod 5738 . . 3 PProd (R, S) = ((R 1st ) ⊗ (S 2nd ))
21breqi 4645 . 2 (A PProd (R, S)BA((R 1st ) ⊗ (S 2nd ))B)
3 brtxp 5783 . 2 (A((R 1st ) ⊗ (S 2nd ))Bzw(B = z, w A(R 1st )z A(S 2nd )w))
4 brco 4883 . . . . . . . 8 (A(R 1st )zx(A1st x xRz))
54anbi1i 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
98br1st 4858 . . . . . . . . . . . 12 (A1st xy A = x, y)
109anbi1i 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 )wx, y(S 2nd )w))
13 vex 2862 . . . . . . . . . . . . . . 15 y V
148, 13brco2nd 5778 . . . . . . . . . . . . . 14 (x, y(S 2nd )wySw)
1512, 14syl6bb 252 . . . . . . . . . . . . 13 (A = x, y → (A(S 2nd )wySw))
1615pm5.32i 618 . . . . . . . . . . . 12 ((A = x, y A(S 2nd )w) ↔ (A = x, y ySw))
1716exbii 1582 . . . . . . . . . . 11 (y(A = x, y A(S 2nd )w) ↔ y(A = x, y ySw))
1810, 11, 173bitr2i 264 . . . . . . . . . 10 ((A1st x A(S 2nd )w) ↔ y(A = x, y ySw))
1918anbi1i 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))
2220, 21bitr3i 242 . . . . . . . . . . 11 ((A = x, y (xRz ySw)) ↔ ((A = x, y ySw) xRz))
2322exbii 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))
2523, 24bitr2i 241 . . . . . . . . 9 ((y(A = x, y ySw) xRz) ↔ y(A = x, y (xRz ySw)))
267, 19, 253bitri 262 . . . . . . . 8 (((A1st x xRz) A(S 2nd )w) ↔ y(A = x, y (xRz ySw)))
2726exbii 1582 . . . . . . 7 (x((A1st x xRz) A(S 2nd )w) ↔ xy(A = x, y (xRz ySw)))
285, 6, 273bitr2i 264 . . . . . 6 ((A(R 1st )z A(S 2nd )w) ↔ xy(A = x, y (xRz ySw)))
2928anbi2i 675 . . . . 5 ((B = z, w (A(R 1st )z A(S 2nd )w)) ↔ (B = z, w xy(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))))
3331, 32bitri 240 . . . . . . 7 ((A = x, y B = z, w (xRz ySw)) ↔ (B = z, w (A = x, y (xRz ySw))))
34332exbii 1583 . . . . . 6 (xy(A = x, y B = z, w (xRz ySw)) ↔ xy(B = z, w (A = x, y (xRz ySw))))
35 19.42vv 1907 . . . . . 6 (xy(B = z, w (A = x, y (xRz ySw))) ↔ (B = z, w xy(A = x, y (xRz ySw))))
3634, 35bitri 240 . . . . 5 (xy(A = x, y B = z, w (xRz ySw)) ↔ (B = z, w xy(A = x, y (xRz ySw))))
3729, 30, 363bitr4i 268 . . . 4 ((B = z, w A(R 1st )z A(S 2nd )w) ↔ xy(A = x, y B = z, w (xRz ySw)))
38372exbii 1583 . . 3 (zw(B = z, w A(R 1st )z A(S 2nd )w) ↔ zwxy(A = x, y B = z, w (xRz ySw)))
39 exrot4 1745 . . 3 (zwxy(A = x, y B = z, w (xRz ySw)) ↔ xyzw(A = x, y B = z, w (xRz ySw)))
4038, 39bitri 240 . 2 (zw(B = z, w A(R 1st )z A(S 2nd )w) ↔ xyzw(A = x, y B = z, w (xRz ySw)))
412, 3, 403bitri 262 1 (A PProd (R, S)Bxyzw(A = x, y B = z, w (xRz ySw)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   ∘ ccom 4721  2nd c2nd 4783   ⊗ ctxp 5735   PProd cpprod 5737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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