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Theorem csbfv12gALT 5552
Description: Move class substitution in and out of a function value.(This is csbfv12g 5551 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbfv12gALTVD 28991. 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 3851 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } } )
2 csbabg 3155 . . . . 5  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  [. A  /  x ]. ( F " { B }
)  =  { y } } )
3 sbceqg 3110 . . . . . . 7  |-  ( A  e.  C  ->  ( [. A  /  x ]. ( F " { B } )  =  {
y }  <->  [_ A  /  x ]_ ( F " { B } )  = 
[_ A  /  x ]_ { y } ) )
4 csbima12g 5038 . . . . . . . . 9  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ { B } ) )
5 csbsng 3705 . . . . . . . . . 10  |-  ( A  e.  C  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
65imaeq2d 5028 . . . . . . . . 9  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
74, 6eqtrd 2328 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
8 csbconstg 3108 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y }  =  { y } )
97, 8eqeq12d 2310 . . . . . . 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 2410 . . . . 5  |-  ( A  e.  C  ->  { y  |  [. A  /  x ]. ( F " { B } )  =  { y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
122, 11eqtrd 2328 . . . 4  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
1312unieqd 3854 . . 3  |-  ( A  e.  C  ->  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }
)
141, 13eqtrd 2328 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
15 dffv4 5538 . . 3  |-  ( F `
 B )  = 
U. { y  |  ( F " { B } )  =  {
y } }
1615csbeq2i 3120 . 2  |-  [_ A  /  x ]_ ( F `
 B )  = 
[_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }
17 dffv4 5538 . 2  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }
1814, 16, 173eqtr4g 2353 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 1632    e. wcel 1696   {cab 2282   [.wsbc 3004   [_csb 3094   {csn 3653   U.cuni 3843   "cima 4708   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279
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