MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1ssres Structured version   Unicode version

Theorem f1ssres 5649
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 5642 . . 3  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fssres 5613 . . 3  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
31, 2sylan 459 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C --> B )
4 df-f1 5462 . . . . 5  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
54simprbi 452 . . . 4  |-  ( F : A -1-1-> B  ->  Fun  `' F )
6 funres11 5524 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  C ) )
75, 6syl 16 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' ( F  |`  C ) )
87adantr 453 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  Fun  `' ( F  |`  C ) )
9 df-f1 5462 . 2  |-  ( ( F  |`  C ) : C -1-1-> B  <->  ( ( F  |`  C ) : C --> B  /\  Fun  `' ( F  |`  C )
) )
103, 8, 9sylanbrc 647 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 360    C_ wss 3322   `'ccnv 4880    |` cres 4883   Fun wfun 5451   -->wf 5453   -1-1->wf1 5454
This theorem is referenced by:  f1ores  5692  oacomf1olem  6810  pwfseqlem5  8543  hashf1lem2  11710  conjsubgen  15043  sylow1lem2  15238  sylow2blem1  15259  usgrares  21394  usgrares1  21429  hashimarn  28186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462
  Copyright terms: Public domain W3C validator