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Theorem f1ssr 5041
Description: Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Assertion
Ref Expression
f1ssr  F : -1-1->  ran  F  C_  C  F : -1-1-> C

Proof of Theorem f1ssr
StepHypRef Expression
1 f1fn 5036 . . . 4  F : -1-1->  F  Fn
21adantr 261 . . 3  F : -1-1->  ran  F  C_  C  F  Fn
3 simpr 103 . . 3  F : -1-1->  ran  F  C_  C  ran  F  C_  C
4 df-f 4849 . . 3  F : --> C  F  Fn  ran  F 
C_  C
52, 3, 4sylanbrc 394 . 2  F : -1-1->  ran  F  C_  C  F : --> C
6 df-f1 4850 . . . 4  F : -1-1->  F : -->  Fun  `' F
76simprbi 260 . . 3  F : -1-1->  Fun  `' F
87adantr 261 . 2  F : -1-1->  ran  F  C_  C  Fun  `' F
9 df-f1 4850 . 2  F : -1-1-> C  F : --> C  Fun  `' F
105, 8, 9sylanbrc 394 1  F : -1-1->  ran  F  C_  C  F : -1-1-> C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97    C_ wss 2911   `'ccnv 4287   ran crn 4289   Fun wfun 4839    Fn wfn 4840   -->wf 4841   -1-1->wf1 4842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110  df-f 4849  df-f1 4850
This theorem is referenced by: (None)
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