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Theorem sbcfg 5125
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 4985 . . . 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 2886 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. F : A --> B  <->  [. X  /  x ]. ( F  Fn  A  /\  ran  F  C_  B ) ) )
4 sbcfng 5124 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
5 sbcssg 3378 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. ran  F  C_  B  <->  [_ X  /  x ]_ ran  F  C_  [_ X  /  x ]_ B ) )
6 csbrng 4858 . . . . . 6  |-  ( X  e.  V  ->  [_ X  /  x ]_ ran  F  =  ran  [_ X  /  x ]_ F )
76sseq1d 3042 . . . . 5  |-  ( X  e.  V  ->  ( [_ X  /  x ]_ ran  F  C_  [_ X  /  x ]_ B  <->  ran  [_ X  /  x ]_ F  C_  [_ X  /  x ]_ B ) )
85, 7bitrd 186 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. ran  F  C_  B  <->  ran  [_ X  /  x ]_ F  C_  [_ X  /  x ]_ B ) )
94, 8anbi12d 457 . . 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 2870 . . 3  |-  ( [. X  /  x ]. ( F  Fn  A  /\  ran  F  C_  B )  <->  (
[. X  /  x ]. F  Fn  A  /\  [. X  /  x ]. ran  F  C_  B
) )
11 df-f 4985 . . 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 221 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. ( F  Fn  A  /\  ran  F  C_  B
)  <->  [_ X  /  x ]_ F : [_ X  /  x ]_ A --> [_ X  /  x ]_ B ) )
133, 12bitrd 186 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 102    <-> wb 103    e. wcel 1436   [.wsbc 2829   [_csb 2922    C_ wss 2988   ran crn 4412    Fn wfn 4976   -->wf 4977
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-id 4094  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-fun 4983  df-fn 4984  df-f 4985
This theorem is referenced by: (None)
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