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Theorem sbcfng 5365
Description: Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
sbcfng  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
Distinct variable groups:    x, V    x, X
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem sbcfng
StepHypRef Expression
1 df-fn 5221 . . . 4  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
21a1i 9 . . 3  |-  ( X  e.  V  ->  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) ) )
32sbcbidv 3023 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [. X  /  x ]. ( Fun  F  /\  dom  F  =  A ) ) )
4 sbcfung 5242 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. Fun  F  <->  Fun  [_ X  /  x ]_ F ) )
5 sbceqg 3075 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. dom  F  =  A  <->  [_ X  /  x ]_ dom  F  =  [_ X  /  x ]_ A
) )
6 csbdmg 4823 . . . . . 6  |-  ( X  e.  V  ->  [_ X  /  x ]_ dom  F  =  dom  [_ X  /  x ]_ F )
76eqeq1d 2186 . . . . 5  |-  ( X  e.  V  ->  ( [_ X  /  x ]_ dom  F  =  [_ X  /  x ]_ A  <->  dom  [_ X  /  x ]_ F  =  [_ X  /  x ]_ A ) )
85, 7bitrd 188 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. dom  F  =  A  <->  dom  [_ X  /  x ]_ F  =  [_ X  /  x ]_ A ) )
94, 8anbi12d 473 . . 3  |-  ( X  e.  V  ->  (
( [. X  /  x ]. Fun  F  /\  [. X  /  x ]. dom  F  =  A )  <->  ( Fun  [_ X  /  x ]_ F  /\  dom  [_ X  /  x ]_ F  = 
[_ X  /  x ]_ A ) ) )
10 sbcan 3007 . . 3  |-  ( [. X  /  x ]. ( Fun  F  /\  dom  F  =  A )  <->  ( [. X  /  x ]. Fun  F  /\  [. X  /  x ]. dom  F  =  A ) )
11 df-fn 5221 . . 3  |-  ( [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A  <->  ( Fun  [_ X  /  x ]_ F  /\  dom  [_ X  /  x ]_ F  =  [_ X  /  x ]_ A ) )
129, 10, 113bitr4g 223 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. ( Fun  F  /\  dom  F  =  A )  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
133, 12bitrd 188 1  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   [.wsbc 2964   [_csb 3059   dom cdm 4628   Fun wfun 5212    Fn wfn 5213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-fun 5220  df-fn 5221
This theorem is referenced by:  sbcfg  5366
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