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Theorem f1ss 5304
Description: A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)
Assertion
Ref Expression
f1ss  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A -1-1-> C )

Proof of Theorem f1ss
StepHypRef Expression
1 f1f 5298 . . 3  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fss 5254 . . 3  |-  ( ( F : A --> B  /\  B  C_  C )  ->  F : A --> C )
31, 2sylan 281 . 2  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A --> C )
4 df-f1 5098 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
54simprbi 273 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' F )
65adantr 274 . 2  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  Fun  `' F
)
7 df-f1 5098 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
83, 6, 7sylanbrc 413 1  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A -1-1-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    C_ wss 3041   `'ccnv 4508   Fun wfun 5087   -->wf 5089   -1-1->wf1 5090
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-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054  df-f 5097  df-f1 5098
This theorem is referenced by:  f1sng  5377
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