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Theorem fss 5373
Description: Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fss  |-  ( ( F : A --> B  /\  B  C_  C )  ->  F : A --> C )

Proof of Theorem fss
StepHypRef Expression
1 sstr2 3162 . . . . 5  |-  ( ran 
F  C_  B  ->  ( B  C_  C  ->  ran 
F  C_  C )
)
21com12 30 . . . 4  |-  ( B 
C_  C  ->  ( ran  F  C_  B  ->  ran 
F  C_  C )
)
32anim2d 337 . . 3  |-  ( B 
C_  C  ->  (
( F  Fn  A  /\  ran  F  C_  B
)  ->  ( F  Fn  A  /\  ran  F  C_  C ) ) )
4 df-f 5216 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
5 df-f 5216 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
63, 4, 53imtr4g 205 . 2  |-  ( B 
C_  C  ->  ( F : A --> B  ->  F : A --> C ) )
76impcom 125 1  |-  ( ( F : A --> B  /\  B  C_  C )  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    C_ wss 3129   ran crn 4624    Fn wfn 5207   -->wf 5208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142  df-f 5216
This theorem is referenced by:  fssd  5374  fconst6g  5410  f1ss  5423  ffoss  5489  fsn2  5686  ofco  6095  tposf2  6263  issmo2  6284  smoiso  6297  mapsn  6684  ssdomg  6772  omp1eomlem  7087  1fv  10122  fxnn0nninf  10421  abscn2  11304  recn2  11306  imcn2  11307  climabs  11309  climre  11311  climim  11312  fsumre  11461  fsumim  11462  ismet2  13514  dvfre  13834  dvrecap  13837  lgsfcl  14069
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