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Theorem sbcfng 5159
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 5018 . . . 4  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
21a1i 9 . . 3  |-  ( X  e.  V  ->  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) ) )
32sbcbidv 2897 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [. X  /  x ]. ( Fun  F  /\  dom  F  =  A ) ) )
4 sbcfung 5039 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. Fun  F  <->  Fun  [_ X  /  x ]_ F ) )
5 sbceqg 2947 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. dom  F  =  A  <->  [_ X  /  x ]_ dom  F  =  [_ X  /  x ]_ A
) )
6 csbdmg 4630 . . . . . 6  |-  ( X  e.  V  ->  [_ X  /  x ]_ dom  F  =  dom  [_ X  /  x ]_ F )
76eqeq1d 2096 . . . . 5  |-  ( X  e.  V  ->  ( [_ X  /  x ]_ dom  F  =  [_ X  /  x ]_ A  <->  dom  [_ X  /  x ]_ F  =  [_ X  /  x ]_ A ) )
85, 7bitrd 186 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. dom  F  =  A  <->  dom  [_ X  /  x ]_ F  =  [_ X  /  x ]_ A ) )
94, 8anbi12d 457 . . 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 2881 . . 3  |-  ( [. X  /  x ]. ( Fun  F  /\  dom  F  =  A )  <->  ( [. X  /  x ]. Fun  F  /\  [. X  /  x ]. dom  F  =  A ) )
11 df-fn 5018 . . 3  |-  ( [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A  <->  ( Fun  [_ X  /  x ]_ F  /\  dom  [_ X  /  x ]_ F  =  [_ X  /  x ]_ A ) )
129, 10, 113bitr4g 221 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. ( Fun  F  /\  dom  F  =  A )  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
133, 12bitrd 186 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   [.wsbc 2840   [_csb 2933   dom cdm 4438   Fun wfun 5009    Fn wfn 5010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-fun 5017  df-fn 5018
This theorem is referenced by:  sbcfg  5160
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