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Theorem csbfv12gALTOLD 31414
Description: Move class substitution in and out of a function value. The proof is derived from the virtual deduction proof csbfv12gALTVD 31492. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 5718 instead. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbfv12gALTOLD  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )

Proof of Theorem csbfv12gALTOLD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbunigOLD 4113 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } } )
2 csbabgOLD 3701 . . . . 5  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  [. A  /  x ]. ( F " { B }
)  =  { y } } )
3 sbceqg 3670 . . . . . . 7  |-  ( A  e.  C  ->  ( [. A  /  x ]. ( F " { B } )  =  {
y }  <->  [_ A  /  x ]_ ( F " { B } )  = 
[_ A  /  x ]_ { y } ) )
4 csbima12gOLD 5180 . . . . . . . . 9  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ { B } ) )
5 csbsng 3928 . . . . . . . . . 10  |-  ( A  e.  C  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
65imaeq2d 5162 . . . . . . . . 9  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
74, 6eqtrd 2469 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
8 csbconstg 3294 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y }  =  { y } )
97, 8eqeq12d 2451 . . . . . . 7  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ ( F " { B } )  =  [_ A  /  x ]_ {
y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) )
103, 9bitrd 253 . . . . . 6  |-  ( A  e.  C  ->  ( [. A  /  x ]. ( F " { B } )  =  {
y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) )
1110abbidv 2551 . . . . 5  |-  ( A  e.  C  ->  { y  |  [. A  /  x ]. ( F " { B } )  =  { y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
122, 11eqtrd 2469 . . . 4  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
1312unieqd 4094 . . 3  |-  ( A  e.  C  ->  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }
)
141, 13eqtrd 2469 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
15 dffv4 5681 . . 3  |-  ( F `
 B )  = 
U. { y  |  ( F " { B } )  =  {
y } }
1615csbeq2i 3681 . 2  |-  [_ A  /  x ]_ ( F `
 B )  = 
[_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }
17 dffv4 5681 . 2  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }
1814, 16, 173eqtr4g 2494 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {cab 2423   [.wsbc 3179   [_csb 3281   {csn 3870   U.cuni 4084   "cima 4835   ` cfv 5411
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524
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-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-br 4286  df-opab 4344  df-xp 4838  df-cnv 4840  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fv 5419
This theorem is referenced by: (None)
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