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Theorem sbcfng 5726
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 5589 . . . 4  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
21a1i 11 . . 3  |-  ( X  e.  V  ->  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) ) )
32sbcbidv 3390 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [. X  /  x ]. ( Fun  F  /\  dom  F  =  A ) ) )
4 sbcfung 5609 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. Fun  F  <->  Fun  [_ X  /  x ]_ F ) )
5 sbceqg 3825 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. dom  F  =  A  <->  [_ X  /  x ]_ dom  F  =  [_ X  /  x ]_ A
) )
6 csbdm 5195 . . . . . 6  |-  [_ X  /  x ]_ dom  F  =  dom  [_ X  /  x ]_ F
76eqeq1i 2474 . . . . 5  |-  ( [_ X  /  x ]_ dom  F  =  [_ X  /  x ]_ A  <->  dom  [_ X  /  x ]_ F  = 
[_ X  /  x ]_ A )
85, 7syl6bb 261 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. dom  F  =  A  <->  dom  [_ X  /  x ]_ F  =  [_ X  /  x ]_ A ) )
94, 8anbi12d 710 . . 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 3374 . . 3  |-  ( [. X  /  x ]. ( Fun  F  /\  dom  F  =  A )  <->  ( [. X  /  x ]. Fun  F  /\  [. X  /  x ]. dom  F  =  A ) )
11 df-fn 5589 . . 3  |-  ( [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A  <->  ( Fun  [_ X  /  x ]_ F  /\  dom  [_ X  /  x ]_ F  =  [_ X  /  x ]_ A ) )
129, 10, 113bitr4g 288 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. ( Fun  F  /\  dom  F  =  A )  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
133, 12bitrd 253 1  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   [.wsbc 3331   [_csb 3435   dom cdm 4999   Fun wfun 5580    Fn wfn 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-fun 5588  df-fn 5589
This theorem is referenced by:  sbcfg  5727
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