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Theorem sbcfng 5636
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 5499 . . . 4  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
21a1i 11 . . 3  |-  ( X  e.  V  ->  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) ) )
32sbcbidv 3307 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [. X  /  x ]. ( Fun  F  /\  dom  F  =  A ) ) )
4 sbcfung 5519 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. Fun  F  <->  Fun  [_ X  /  x ]_ F ) )
5 sbceqg 3751 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. dom  F  =  A  <->  [_ X  /  x ]_ dom  F  =  [_ X  /  x ]_ A
) )
6 csbdm 5110 . . . . . 6  |-  [_ X  /  x ]_ dom  F  =  dom  [_ X  /  x ]_ F
76eqeq1i 2389 . . . . 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 708 . . 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 3295 . . 3  |-  ( [. X  /  x ]. ( Fun  F  /\  dom  F  =  A )  <->  ( [. X  /  x ]. Fun  F  /\  [. X  /  x ]. dom  F  =  A ) )
11 df-fn 5499 . . 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 367    = wceq 1399    e. wcel 1826   [.wsbc 3252   [_csb 3348   dom cdm 4913   Fun wfun 5490    Fn wfn 5491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-id 4709  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-fun 5498  df-fn 5499
This theorem is referenced by:  sbcfg  5637
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