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Theorem f1ssr 5787
Description: A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Assertion
Ref Expression
f1ssr  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A -1-1-> C )

Proof of Theorem f1ssr
StepHypRef Expression
1 f1fn 5782 . . . 4  |-  ( F : A -1-1-> B  ->  F  Fn  A )
21adantr 465 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F  Fn  A )
3 simpr 461 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  ran  F  C_  C )
4 df-f 5592 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
52, 3, 4sylanbrc 664 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A
--> C )
6 df-f1 5593 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
76simprbi 464 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' F )
87adantr 465 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  Fun  `' F
)
9 df-f1 5593 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
105, 8, 9sylanbrc 664 1  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A -1-1-> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    C_ wss 3476   `'ccnv 4998   ran crn 5000   Fun wfun 5582    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-f 5592  df-f1 5593
This theorem is referenced by:  domdifsn  7600  marypha1  7894  m2cpmf1  19039  usgrares1  24114  usgresvm1  31938  usgresvm1ALT  31942
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