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Theorem csbfv12 5916
Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbfv12  |-  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )

Proof of Theorem csbfv12
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbiota 5594 . . . 4  |-  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [. A  /  x ]. B F y )
2 sbcbr123 4477 . . . . . 6  |-  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y
)
3 csbconstg 3414 . . . . . . 7  |-  ( A  e.  _V  ->  [_ A  /  x ]_ y  =  y )
43breq2d 4438 . . . . . 6  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
52, 4syl5bb 260 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
65iotabidv 5586 . . . 4  |-  ( A  e.  _V  ->  ( iota y [. A  /  x ]. B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
71, 6syl5eq 2482 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
8 df-fv 5609 . . . 4  |-  ( F `
 B )  =  ( iota y B F y )
98csbeq2i 3816 . . 3  |-  [_ A  /  x ]_ ( F `
 B )  = 
[_ A  /  x ]_ ( iota y B F y )
10 df-fv 5609 . . 3  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F
y )
117, 9, 103eqtr4g 2495 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
12 csbprc 3804 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F `  B )  =  (/) )
13 csbprc 3804 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ F  =  (/) )
1413fveq1d 5883 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( (/) `  [_ A  /  x ]_ B ) )
15 0fv 5914 . . . 4  |-  ( (/) ` 
[_ A  /  x ]_ B )  =  (/)
1614, 15syl6req 2487 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
1712, 16eqtrd 2470 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
1811, 17pm2.61i 167 1  |-  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437    e. wcel 1870   _Vcvv 3087   [.wsbc 3305   [_csb 3401   (/)c0 3767   class class class wbr 4426   iotacio 5563   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-nul 4556  ax-pow 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-dm 4864  df-iota 5565  df-fv 5609
This theorem is referenced by:  csbfv2g  5917  coe1fzgsumdlem  18830  evl1gsumdlem  18879  cdlemk42  34220  iccelpart  38149
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