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Theorem f1ssres 5609
Description: A function that is one-to-one is also one-to-one on some aubset 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 5602 . . 3  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fssres 5573 . . 3  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
31, 2sylan 458 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C --> B )
4 df-f1 5422 . . . . 5  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
54simprbi 451 . . . 4  |-  ( F : A -1-1-> B  ->  Fun  `' F )
6 funres11 5484 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  C ) )
75, 6syl 16 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' ( F  |`  C ) )
87adantr 452 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  Fun  `' ( F  |`  C ) )
9 df-f1 5422 . 2  |-  ( ( F  |`  C ) : C -1-1-> B  <->  ( ( F  |`  C ) : C --> B  /\  Fun  `' ( F  |`  C )
) )
103, 8, 9sylanbrc 646 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 359    C_ wss 3284   `'ccnv 4840    |` cres 4843   Fun wfun 5411   -->wf 5413   -1-1->wf1 5414
This theorem is referenced by:  f1ores  5652  oacomf1olem  6770  pwfseqlem5  8498  hashf1lem2  11664  conjsubgen  14997  sylow1lem2  15192  sylow2blem1  15213  usgrares  21346  usgrares1  21381  hashimarn  27998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422
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