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Theorem csbfv12gALTOLD 32576
Description: Move class substitution in and out of a function value. The proof is derived from the virtual deduction proof csbfv12gALTVD 32654. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 5892 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 4267 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } } )
2 csbabgOLD 3849 . . . . 5  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  [. A  /  x ]. ( F " { B }
)  =  { y } } )
3 sbceqg 3818 . . . . . . 7  |-  ( A  e.  C  ->  ( [. A  /  x ]. ( F " { B } )  =  {
y }  <->  [_ A  /  x ]_ ( F " { B } )  = 
[_ A  /  x ]_ { y } ) )
4 csbima12gOLD 5346 . . . . . . . . 9  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ { B } ) )
5 csbsng 4079 . . . . . . . . . 10  |-  ( A  e.  C  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
65imaeq2d 5328 . . . . . . . . 9  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
74, 6eqtrd 2501 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
8 csbconstg 3441 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y }  =  { y } )
97, 8eqeq12d 2482 . . . . . . 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 2596 . . . . 5  |-  ( A  e.  C  ->  { y  |  [. A  /  x ]. ( F " { B } )  =  { y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
122, 11eqtrd 2501 . . . 4  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
1312unieqd 4248 . . 3  |-  ( A  e.  C  ->  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }
)
141, 13eqtrd 2501 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
15 dffv4 5854 . . 3  |-  ( F `
 B )  = 
U. { y  |  ( F " { B } )  =  {
y } }
1615csbeq2i 3829 . 2  |-  [_ A  /  x ]_ ( F `
 B )  = 
[_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }
17 dffv4 5854 . 2  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }
1814, 16, 173eqtr4g 2526 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 1374    e. wcel 1762   {cab 2445   [.wsbc 3324   [_csb 3428   {csn 4020   U.cuni 4238   "cima 4995   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fv 5587
This theorem is referenced by: (None)
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