New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  brtxp GIF version

Theorem brtxp 5783
 Description: Binary relationship over a tail cross product. (Contributed by SF, 11-Feb-2015.)
Assertion
Ref Expression
brtxp (A(RS)Bxy(B = x, y ARx ASy))
Distinct variable groups:   x,A,y   x,B,y   x,R,y   x,S,y

Proof of Theorem brtxp
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brin 4693 . . . 4 (A((1st R) ∩ (2nd S))B ↔ (A(1st R)B A(2nd S)B))
2 brco 4883 . . . . 5 (A(1st R)Bx(ARx x1st B))
3 brco 4883 . . . . 5 (A(2nd S)By(ASy y2nd B))
42, 3anbi12i 678 . . . 4 ((A(1st R)B A(2nd S)B) ↔ (x(ARx x1st B) y(ASy y2nd B)))
51, 4bitri 240 . . 3 (A((1st R) ∩ (2nd S))B ↔ (x(ARx x1st B) y(ASy y2nd B)))
6 df-txp 5736 . . . 4 (RS) = ((1st R) ∩ (2nd S))
76breqi 4645 . . 3 (A(RS)BA((1st R) ∩ (2nd S))B)
8 eeanv 1913 . . 3 (xy((ARx x1st B) (ASy y2nd B)) ↔ (x(ARx x1st B) y(ASy y2nd B)))
95, 7, 83bitr4i 268 . 2 (A(RS)Bxy((ARx x1st B) (ASy y2nd B)))
10 an4 797 . . . 4 (((ARx x1st B) (ASy y2nd B)) ↔ ((ARx ASy) (x1st B y2nd B)))
11 ancom 437 . . . . 5 (((ARx ASy) B = x, y) ↔ (B = x, y (ARx ASy)))
12 brcnv 4892 . . . . . . . . 9 (x1st BB1st x)
13 vex 2862 . . . . . . . . . 10 x V
1413br1st 4858 . . . . . . . . 9 (B1st xz B = x, z)
1512, 14bitri 240 . . . . . . . 8 (x1st Bz B = x, z)
16 brcnv 4892 . . . . . . . . 9 (y2nd BB2nd y)
17 vex 2862 . . . . . . . . . 10 y V
1817br2nd 4859 . . . . . . . . 9 (B2nd yw B = w, y)
1916, 18bitri 240 . . . . . . . 8 (y2nd Bw B = w, y)
2015, 19anbi12i 678 . . . . . . 7 ((x1st B y2nd B) ↔ (z B = x, z w B = w, y))
21 eeanv 1913 . . . . . . 7 (zw(B = x, z B = w, y) ↔ (z B = x, z w B = w, y))
22 eqtr2 2371 . . . . . . . . . . 11 ((B = x, z B = w, y) → x, z = w, y)
23 opth 4602 . . . . . . . . . . . . . 14 (x, z = w, y ↔ (x = w z = y))
2423simplbi 446 . . . . . . . . . . . . 13 (x, z = w, yx = w)
2524eqcomd 2358 . . . . . . . . . . . 12 (x, z = w, yw = x)
2625opeq1d 4584 . . . . . . . . . . 11 (x, z = w, yw, y = x, y)
2722, 26syl 15 . . . . . . . . . 10 ((B = x, z B = w, y) → w, y = x, y)
28 eqeq1 2359 . . . . . . . . . . 11 (B = w, y → (B = x, yw, y = x, y))
2928adantl 452 . . . . . . . . . 10 ((B = x, z B = w, y) → (B = x, yw, y = x, y))
3027, 29mpbird 223 . . . . . . . . 9 ((B = x, z B = w, y) → B = x, y)
3130exlimivv 1635 . . . . . . . 8 (zw(B = x, z B = w, y) → B = x, y)
32 opeq2 4579 . . . . . . . . . . . 12 (z = yx, z = x, y)
3332eqeq2d 2364 . . . . . . . . . . 11 (z = y → (B = x, zB = x, y))
34 opeq1 4578 . . . . . . . . . . . 12 (w = xw, y = x, y)
3534eqeq2d 2364 . . . . . . . . . . 11 (w = x → (B = w, yB = x, y))
3633, 35bi2anan9 843 . . . . . . . . . 10 ((z = y w = x) → ((B = x, z B = w, y) ↔ (B = x, y B = x, y)))
3717, 13, 36spc2ev 2947 . . . . . . . . 9 ((B = x, y B = x, y) → zw(B = x, z B = w, y))
3837anidms 626 . . . . . . . 8 (B = x, yzw(B = x, z B = w, y))
3931, 38impbii 180 . . . . . . 7 (zw(B = x, z B = w, y) ↔ B = x, y)
4020, 21, 393bitr2i 264 . . . . . 6 ((x1st B y2nd B) ↔ B = x, y)
4140anbi2i 675 . . . . 5 (((ARx ASy) (x1st B y2nd B)) ↔ ((ARx ASy) B = x, y))
42 3anass 938 . . . . 5 ((B = x, y ARx ASy) ↔ (B = x, y (ARx ASy)))
4311, 41, 423bitr4i 268 . . . 4 (((ARx ASy) (x1st B y2nd B)) ↔ (B = x, y ARx ASy))
4410, 43bitri 240 . . 3 (((ARx x1st B) (ASy y2nd B)) ↔ (B = x, y ARx ASy))
45442exbii 1583 . 2 (xy((ARx x1st B) (ASy y2nd B)) ↔ xy(B = x, y ARx ASy))
469, 45bitri 240 1 (A(RS)Bxy(B = x, y ARx ASy))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∩ cin 3208  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   ∘ ccom 4721  ◡ccnv 4771  2nd c2nd 4783   ⊗ ctxp 5735 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 This theorem is referenced by:  restxp  5786  oqelins4  5794  dmtxp  5802  fntxp  5804  brpprod  5839
 Copyright terms: Public domain W3C validator