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Theorem csbcnvgALT 5010
Description: Move class substitution in and out of the converse of a function. Version of csbcnv 5009 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 4448 . . . . 5  |-  ( [. A  /  x ]. z F y  <->  [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y
)
2 csbconstg 3388 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ z  =  z )
3 csbconstg 3388 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
42, 3breq12d 4410 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ z [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  z [_ A  /  x ]_ F
y ) )
51, 4syl5bb 259 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. z F y  <->  z [_ A  /  x ]_ F
y ) )
65opabbidv 4460 . . 3  |-  ( A  e.  V  ->  { <. y ,  z >.  |  [. A  /  x ]. z F y }  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y } )
7 csbopabgALT 4725 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  z F y }  =  { <. y ,  z
>.  |  [. A  /  x ]. z F y } )
8 df-cnv 4833 . . . 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 2456 . 2  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ { <. y ,  z >.  |  z F y } )
11 df-cnv 4833 . . 3  |-  `' F  =  { <. y ,  z
>.  |  z F
y }
1211csbeq2i 3789 . 2  |-  [_ A  /  x ]_ `' F  =  [_ A  /  x ]_ { <. y ,  z
>.  |  z F
y }
1310, 12syl6eqr 2463 1  |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844   [.wsbc 3279   [_csb 3375   class class class wbr 4397   {copab 4454   `'ccnv 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-cnv 4833
This theorem is referenced by: (None)
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