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Theorem f1ssres 5332
Description: A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
Assertion
Ref Expression
f1ssres  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )

Proof of Theorem f1ssres
StepHypRef Expression
1 f1f 5323 . . 3  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fssres 5293 . . 3  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
31, 2sylan 281 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C --> B )
4 df-f1 5123 . . . . 5  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
54simprbi 273 . . . 4  |-  ( F : A -1-1-> B  ->  Fun  `' F )
6 funres11 5190 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  C ) )
75, 6syl 14 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' ( F  |`  C ) )
87adantr 274 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  Fun  `' ( F  |`  C ) )
9 df-f1 5123 . 2  |-  ( ( F  |`  C ) : C -1-1-> B  <->  ( ( F  |`  C ) : C --> B  /\  Fun  `' ( F  |`  C )
) )
103, 8, 9sylanbrc 413 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    C_ wss 3066   `'ccnv 4533    |` cres 4536   Fun wfun 5112   -->wf 5114   -1-1->wf1 5115
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123
This theorem is referenced by:  f1resf1  5333  f1ores  5375
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