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

Theorem brco 4883
 Description: Binary relation on a composition. (Contributed by set.mm contributors, 21-Sep-2004.)
Assertion
Ref Expression
brco (A(C D)Bx(ADx xCB))
Distinct variable groups:   x,A   x,B   x,C   x,D

Proof of Theorem brco
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brex 4689 . 2 (A(C D)B → (A V B V))
2 brex 4689 . . . . 5 (ADx → (A V x V))
32simpld 445 . . . 4 (ADxA V)
4 brex 4689 . . . . 5 (xCB → (x V B V))
54simprd 449 . . . 4 (xCBB V)
63, 5anim12i 549 . . 3 ((ADx xCB) → (A V B V))
76exlimiv 1634 . 2 (x(ADx xCB) → (A V B V))
8 breq1 4642 . . . . 5 (y = A → (yDxADx))
98anbi1d 685 . . . 4 (y = A → ((yDx xCz) ↔ (ADx xCz)))
109exbidv 1626 . . 3 (y = A → (x(yDx xCz) ↔ x(ADx xCz)))
11 breq2 4643 . . . . 5 (z = B → (xCzxCB))
1211anbi2d 684 . . . 4 (z = B → ((ADx xCz) ↔ (ADx xCB)))
1312exbidv 1626 . . 3 (z = B → (x(ADx xCz) ↔ x(ADx xCB)))
14 df-co 4726 . . 3 (C D) = {y, z x(yDx xCz)}
1510, 13, 14brabg 4706 . 2 ((A V B V) → (A(C D)Bx(ADx xCB)))
161, 7, 15pm5.21nii 342 1 (A(C D)Bx(ADx xCB))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   class class class wbr 4639   ∘ ccom 4721 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-co 4726 This theorem is referenced by:  opelco  4884  cnvco  4894  dmcoss  4971  dmcosseq  4973  cotr  5026  resco  5085  imaco  5086  rnco  5087  coi1  5094  coass  5097  dmfco  5381  brco1st  5777  brco2nd  5778  trtxp  5781  brtxp  5783  addcfnex  5824  qrpprod  5836  brpprod  5839  fnfullfunlem1  5856  clos1ex  5876  xpassenlem  6056  xpassen  6057  enpw1lem1  6061  enmap2lem1  6063  enmap1lem1  6069  lecex  6115  ceex  6174  nnltp1clem1  6261  addccan2nclem1  6263  addccan2nclem2  6264  nncdiv3lem1  6275  nchoicelem10  6298  nchoicelem16  6304
 Copyright terms: Public domain W3C validator