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

Theorem csbcnvg 4793
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 4041 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. z F y  <->  [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y
) )
2 csbconstg 3063 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ z  =  z )
3 csbconstg 3063 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
42, 3breq12d 4000 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  z [_ A  /  x ]_ F
y ) )
51, 4bitrd 187 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. z F y  <->  z [_ A  /  x ]_ F
y ) )
65opabbidv 4053 . . 3  |-  ( A  e.  V  ->  { <. y ,  z >.  |  [. A  /  x ]. z F y }  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y } )
7 csbopabg 4065 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  z F y }  =  { <. y ,  z
>.  |  [. A  /  x ]. z F y } )
8 df-cnv 4617 . . . 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 2214 . 2  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ { <. y ,  z >.  |  z F y } )
11 df-cnv 4617 . . 3  |-  `' F  =  { <. y ,  z
>.  |  z F
y }
1211csbeq2i 3076 . 2  |-  [_ A  /  x ]_ `' F  =  [_ A  /  x ]_ { <. y ,  z
>.  |  z F
y }
1310, 12eqtr4di 2221 1  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   [.wsbc 2955   [_csb 3049   class class class wbr 3987   {copab 4047   `'ccnv 4608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-cnv 4617
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator