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Theorem f1ss 5439
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 5433 . . 3  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fss 5389 . . 3  |-  ( ( F : A --> B  /\  B  C_  C )  ->  F : A --> C )
31, 2sylan 283 . 2  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A --> C )
4 df-f1 5233 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
54simprbi 275 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' F )
65adantr 276 . 2  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  Fun  `' F
)
7 df-f1 5233 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
83, 6, 7sylanbrc 417 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 104    C_ wss 3141   `'ccnv 4637   Fun wfun 5222   -->wf 5224   -1-1->wf1 5225
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-in 3147  df-ss 3154  df-f 5232  df-f1 5233
This theorem is referenced by:  f1sng  5515
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