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Theorem sbcfg 5278
Description: Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
sbcfg  |-  ( X  e.  V  ->  ( [. X  /  x ]. F : A --> B  <->  [_ X  /  x ]_ F : [_ X  /  x ]_ A --> [_ X  /  x ]_ B ) )
Distinct variable groups:    x, V    x, X
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem sbcfg
StepHypRef Expression
1 df-f 5134 . . . 4  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
21a1i 9 . . 3  |-  ( X  e.  V  ->  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) ) )
32sbcbidv 2970 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. F : A --> B  <->  [. X  /  x ]. ( F  Fn  A  /\  ran  F  C_  B ) ) )
4 sbcfng 5277 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
5 sbcssg 3476 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. ran  F  C_  B  <->  [_ X  /  x ]_ ran  F  C_  [_ X  /  x ]_ B ) )
6 csbrng 5007 . . . . . 6  |-  ( X  e.  V  ->  [_ X  /  x ]_ ran  F  =  ran  [_ X  /  x ]_ F )
76sseq1d 3130 . . . . 5  |-  ( X  e.  V  ->  ( [_ X  /  x ]_ ran  F  C_  [_ X  /  x ]_ B  <->  ran  [_ X  /  x ]_ F  C_  [_ X  /  x ]_ B ) )
85, 7bitrd 187 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. ran  F  C_  B  <->  ran  [_ X  /  x ]_ F  C_  [_ X  /  x ]_ B ) )
94, 8anbi12d 465 . . 3  |-  ( X  e.  V  ->  (
( [. X  /  x ]. F  Fn  A  /\  [. X  /  x ]. ran  F  C_  B
)  <->  ( [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A  /\  ran  [_ X  /  x ]_ F  C_  [_ X  /  x ]_ B ) ) )
10 sbcan 2954 . . 3  |-  ( [. X  /  x ]. ( F  Fn  A  /\  ran  F  C_  B )  <->  (
[. X  /  x ]. F  Fn  A  /\  [. X  /  x ]. ran  F  C_  B
) )
11 df-f 5134 . . 3  |-  ( [_ X  /  x ]_ F : [_ X  /  x ]_ A --> [_ X  /  x ]_ B  <->  ( [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A  /\  ran  [_ X  /  x ]_ F  C_  [_ X  /  x ]_ B ) )
129, 10, 113bitr4g 222 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. ( F  Fn  A  /\  ran  F  C_  B
)  <->  [_ X  /  x ]_ F : [_ X  /  x ]_ A --> [_ X  /  x ]_ B ) )
133, 12bitrd 187 1  |-  ( X  e.  V  ->  ( [. X  /  x ]. F : A --> B  <->  [_ X  /  x ]_ F : [_ X  /  x ]_ A --> [_ X  /  x ]_ B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1481   [.wsbc 2912   [_csb 3006    C_ wss 3075   ran crn 4547    Fn wfn 5125   -->wf 5126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-id 4222  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-fun 5132  df-fn 5133  df-f 5134
This theorem is referenced by:  ctiunctlemf  11985
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