MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvco Unicode version

Theorem cnvco 4867
Description: Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvco  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )

Proof of Theorem cnvco
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exancom 1575 . . . 4  |-  ( E. z ( x B z  /\  z A y )  <->  E. z
( z A y  /\  x B z ) )
2 vex 2793 . . . . 5  |-  x  e. 
_V
3 vex 2793 . . . . 5  |-  y  e. 
_V
42, 3brco 4854 . . . 4  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
5 vex 2793 . . . . . . 7  |-  z  e. 
_V
63, 5brcnv 4866 . . . . . 6  |-  ( y `' A z  <->  z A
y )
75, 2brcnv 4866 . . . . . 6  |-  ( z `' B x  <->  x B
z )
86, 7anbi12i 678 . . . . 5  |-  ( ( y `' A z  /\  z `' B x )  <->  ( z A y  /\  x B z ) )
98exbii 1571 . . . 4  |-  ( E. z ( y `' A z  /\  z `' B x )  <->  E. z
( z A y  /\  x B z ) )
101, 4, 93bitr4i 268 . . 3  |-  ( x ( A  o.  B
) y  <->  E. z
( y `' A
z  /\  z `' B x ) )
1110opabbii 4085 . 2  |-  { <. y ,  x >.  |  x ( A  o.  B
) y }  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
12 df-cnv 4699 . 2  |-  `' ( A  o.  B )  =  { <. y ,  x >.  |  x
( A  o.  B
) y }
13 df-co 4700 . 2  |-  ( `' B  o.  `' A
)  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
1411, 12, 133eqtr4i 2315 1  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1530    = wceq 1625   class class class wbr 4025   {copab 4078   `'ccnv 4690    o. ccom 4695
This theorem is referenced by:  rncoss  4947  rncoeq  4950  dmco  5183  cores2  5187  co01  5189  coi2  5191  relcnvtr  5194  dfdm2  5206  f1co  5448  cofunex2g  5742  fparlem3  6222  fparlem4  6223  suppfif1  7151  mapfien  7401  cnvps  14323  gimco  14734  gsumval3  15193  gsumzf1o  15198  cnco  16997  ptrescn  17335  qtopcn  17407  hmeoco  17465  cncombf  19015  deg1val  19484  mbfmco  23571  cvmliftmolem1  23814  cvmlift2lem9a  23836  cvmlift2lem9  23844  relexpcnv  24031  trlcocnv  30982  tendoicl  31058  cdlemk45  31209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-cnv 4699  df-co 4700
  Copyright terms: Public domain W3C validator