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Theorem csbcnvg 4462
Description: Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
Assertion
Ref Expression
csbcnvg  V  `' [_  ]_ F  [_  ]_ `' F

Proof of Theorem csbcnvg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbrg 3804 . . . . 5  V  [.  ]. F  [_  ]_ [_  ]_ F [_  ]_
2 csbconstg 2858 . . . . . 6  V  [_  ]_
3 csbconstg 2858 . . . . . 6  V  [_  ]_
42, 3breq12d 3768 . . . . 5  V  [_  ]_ [_  ]_ F [_  ]_  [_  ]_ F
51, 4bitrd 177 . . . 4  V  [.  ]. F  [_  ]_ F
65opabbidv 3814 . . 3  V  { <. ,  >.  | 
[.  ]. F }  { <. , 
>.  |  [_  ]_ F }
7 csbopabg 3826 . . 3  V  [_  ]_ { <. ,  >.  |  F }  { <. , 
>.  |  [.  ]. F }
8 df-cnv 4296 . . . 4  `' [_  ]_ F  { <. , 
>.  |  [_  ]_ F }
98a1i 9 . . 3  V  `' [_  ]_ F  { <. ,  >.  |  [_  ]_ F }
106, 7, 93eqtr4rd 2080 . 2  V  `' [_  ]_ F  [_  ]_ { <. ,  >.  |  F }
11 df-cnv 4296 . . 3  `' F  { <. , 
>.  |  F }
1211csbeq2i 2870 . 2  [_  ]_ `' F  [_  ]_ { <. , 
>.  |  F }
1310, 12syl6eqr 2087 1  V  `' [_  ]_ F  [_  ]_ `' F
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390   [.wsbc 2758   [_csb 2846   class class class wbr 3755   {copab 3808   `'ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296
This theorem is referenced by: (None)
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