ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbcnvg Unicode version

Theorem csbcnvg 4547
Description: Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
Assertion
Ref Expression
csbcnvg  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )

Proof of Theorem csbcnvg
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbrg 3842 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. z F y  <->  [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y
) )
2 csbconstg 2921 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ z  =  z )
3 csbconstg 2921 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
42, 3breq12d 3806 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  z [_ A  /  x ]_ F
y ) )
51, 4bitrd 186 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. z F y  <->  z [_ A  /  x ]_ F
y ) )
65opabbidv 3852 . . 3  |-  ( A  e.  V  ->  { <. y ,  z >.  |  [. A  /  x ]. z F y }  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y } )
7 csbopabg 3864 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  z F y }  =  { <. y ,  z
>.  |  [. A  /  x ]. z F y } )
8 df-cnv 4379 . . . 4  |-  `' [_ A  /  x ]_ F  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y }
98a1i 9 . . 3  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  { <. y ,  z >.  |  z
[_ A  /  x ]_ F y } )
106, 7, 93eqtr4rd 2125 . 2  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ { <. y ,  z >.  |  z F y } )
11 df-cnv 4379 . . 3  |-  `' F  =  { <. y ,  z
>.  |  z F
y }
1211csbeq2i 2933 . 2  |-  [_ A  /  x ]_ `' F  =  [_ A  /  x ]_ { <. y ,  z
>.  |  z F
y }
1310, 12syl6eqr 2132 1  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   [.wsbc 2816   [_csb 2909   class class class wbr 3793   {copab 3846   `'ccnv 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-cnv 4379
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator