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Theorem csbfv12gALT 5536
Description: Move class substitution in and out of a function value.(This is csbfv12g 5535 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbfv12gALTVD 28675. Although the proof is shorter, the total number of steps of all theorems used in the proof is probably longer. (Contributed by NM, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbfv12gALT  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )

Proof of Theorem csbfv12gALT
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbunig 3835 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } } )
2 csbabg 3142 . . . . 5  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  [. A  /  x ]. ( F " { B }
)  =  { y } } )
3 sbceqg 3097 . . . . . . 7  |-  ( A  e.  C  ->  ( [. A  /  x ]. ( F " { B } )  =  {
y }  <->  [_ A  /  x ]_ ( F " { B } )  = 
[_ A  /  x ]_ { y } ) )
4 csbima12g 5022 . . . . . . . . 9  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ { B } ) )
5 csbsng 3692 . . . . . . . . . 10  |-  ( A  e.  C  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
65imaeq2d 5012 . . . . . . . . 9  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
74, 6eqtrd 2315 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
8 csbconstg 3095 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y }  =  { y } )
97, 8eqeq12d 2297 . . . . . . 7  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ ( F " { B } )  =  [_ A  /  x ]_ {
y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) )
103, 9bitrd 244 . . . . . 6  |-  ( A  e.  C  ->  ( [. A  /  x ]. ( F " { B } )  =  {
y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) )
1110abbidv 2397 . . . . 5  |-  ( A  e.  C  ->  { y  |  [. A  /  x ]. ( F " { B } )  =  { y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
122, 11eqtrd 2315 . . . 4  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
1312unieqd 3838 . . 3  |-  ( A  e.  C  ->  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }
)
141, 13eqtrd 2315 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
15 dffv4 5522 . . 3  |-  ( F `
 B )  = 
U. { y  |  ( F " { B } )  =  {
y } }
1615csbeq2i 3107 . 2  |-  [_ A  /  x ]_ ( F `
 B )  = 
[_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }
17 dffv4 5522 . 2  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }
1814, 16, 173eqtr4g 2340 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   [.wsbc 2991   [_csb 3081   {csn 3640   U.cuni 3827   "cima 4692   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263
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