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Theorem csbcnvgALT 5045
Description: Move class substitution in and out of the converse of a function. Version of csbcnv 5044 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbcnvgALT  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )

Proof of Theorem csbcnvgALT
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbr123 4364 . . . . 5  |-  ( [. A  /  x ]. z F y  <->  [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y
)
2 csbconstg 3322 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ z  =  z )
3 csbconstg 3322 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
42, 3breq12d 4326 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  z [_ A  /  x ]_ F
y ) )
51, 4syl5bb 257 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. z F y  <->  z [_ A  /  x ]_ F
y ) )
65opabbidv 4376 . . 3  |-  ( A  e.  V  ->  { <. y ,  z >.  |  [. A  /  x ]. z F y }  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y } )
7 csbopabgALT 4642 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  z F y }  =  { <. y ,  z
>.  |  [. A  /  x ]. z F y } )
8 df-cnv 4869 . . . 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 2486 . 2  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ { <. y ,  z >.  |  z F y } )
11 df-cnv 4869 . . 3  |-  `' F  =  { <. y ,  z
>.  |  z F
y }
1211csbeq2i 3709 . 2  |-  [_ A  /  x ]_ `' F  =  [_ A  /  x ]_ { <. y ,  z
>.  |  z F
y }
1310, 12syl6eqr 2493 1  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   [.wsbc 3207   [_csb 3309   class class class wbr 4313   {copab 4370   `'ccnv 4860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372  df-cnv 4869
This theorem is referenced by: (None)
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