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Theorem f1ss 5125
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 5120 . . 3  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fss 5082 . . 3  |-  ( ( F : A --> B  /\  B  C_  C )  ->  F : A --> C )
31, 2sylan 271 . 2  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A --> C )
4 df-f1 4935 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
54simprbi 264 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' F )
65adantr 265 . 2  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  Fun  `' F
)
7 df-f1 4935 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
83, 6, 7sylanbrc 402 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 101    C_ wss 2945   `'ccnv 4372   Fun wfun 4924   -->wf 4926   -1-1->wf1 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959  df-f 4934  df-f1 4935
This theorem is referenced by: (None)
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