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Theorem csbcnvg 23990
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 4221 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. z F y  <->  [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y
) )
2 csbconstg 3225 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ z  =  z )
3 csbconstg 3225 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
42, 3breq12d 4185 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  z [_ A  /  x ]_ F
y ) )
51, 4bitrd 245 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. z F y  <->  z [_ A  /  x ]_ F
y ) )
65opabbidv 4231 . . 3  |-  ( A  e.  V  ->  { <. y ,  z >.  |  [. A  /  x ]. z F y }  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y } )
7 csbopabg 4243 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  z F y }  =  { <. y ,  z
>.  |  [. A  /  x ]. z F y } )
8 df-cnv 4845 . . . 4  |-  `' [_ A  /  x ]_ F  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y }
98a1i 11 . . 3  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  { <. y ,  z >.  |  z
[_ A  /  x ]_ F y } )
106, 7, 93eqtr4rd 2447 . 2  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ { <. y ,  z >.  |  z F y } )
11 df-cnv 4845 . . 3  |-  `' F  =  { <. y ,  z
>.  |  z F
y }
1211csbeq2i 3237 . 2  |-  [_ A  /  x ]_ `' F  =  [_ A  /  x ]_ { <. y ,  z
>.  |  z F
y }
1310, 12syl6eqr 2454 1  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   [.wsbc 3121   [_csb 3211   class class class wbr 4172   {copab 4225   `'ccnv 4836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-cnv 4845
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