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Theorem f1ssres 5625
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 5618 . . 3  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fssres 5590 . . 3  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
31, 2sylan 471 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C --> B )
4 df-f1 5435 . . . . 5  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
54simprbi 464 . . . 4  |-  ( F : A -1-1-> B  ->  Fun  `' F )
6 funres11 5498 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  C ) )
75, 6syl 16 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' ( F  |`  C ) )
87adantr 465 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  Fun  `' ( F  |`  C ) )
9 df-f1 5435 . 2  |-  ( ( F  |`  C ) : C -1-1-> B  <->  ( ( F  |`  C ) : C --> B  /\  Fun  `' ( F  |`  C )
) )
103, 8, 9sylanbrc 664 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    C_ wss 3340   `'ccnv 4851    |` cres 4854   Fun wfun 5424   -->wf 5426   -1-1->wf1 5427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-br 4305  df-opab 4363  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435
This theorem is referenced by:  f1ores  5667  oacomf1olem  7015  pwfseqlem5  8842  hashimarn  12212  hashf1lem2  12221  conjsubgen  15791  sylow1lem2  16110  sylow2blem1  16131  usgrares  23300  usgrares1  23335
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