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Theorem sbcfg 5096
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 4956 . . . 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 2881 . 2  |-  ( X  e.  V  ->  ( [. X  /  x ]. F : A --> B  <->  [. X  /  x ]. ( F  Fn  A  /\  ran  F  C_  B ) ) )
4 sbcfng 5095 . . . 4  |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
5 sbcssg 3367 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. ran  F  C_  B  <->  [_ X  /  x ]_ ran  F  C_  [_ X  /  x ]_ B ) )
6 csbrng 4832 . . . . . 6  |-  ( X  e.  V  ->  [_ X  /  x ]_ ran  F  =  ran  [_ X  /  x ]_ F )
76sseq1d 3035 . . . . 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 2865 . . 3  |-  ( [. X  /  x ]. ( F  Fn  A  /\  ran  F  C_  B )  <->  (
[. X  /  x ]. F  Fn  A  /\  [. X  /  x ]. ran  F  C_  B
) )
11 df-f 4956 . . 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 1434   [.wsbc 2824   [_csb 2917    C_ wss 2982   ran crn 4392    Fn wfn 4947   -->wf 4948
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956
This theorem is referenced by: (None)
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