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Theorem f1ssres 5770
Description: A function that is one-to-one is also one-to-one on some subset 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 5763 . . 3  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fssres 5733 . . 3  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
31, 2sylan 469 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C --> B )
4 df-f1 5575 . . . . 5  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
54simprbi 462 . . . 4  |-  ( F : A -1-1-> B  ->  Fun  `' F )
6 funres11 5638 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  C ) )
75, 6syl 16 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' ( F  |`  C ) )
87adantr 463 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  Fun  `' ( F  |`  C ) )
9 df-f1 5575 . 2  |-  ( ( F  |`  C ) : C -1-1-> B  <->  ( ( F  |`  C ) : C --> B  /\  Fun  `' ( F  |`  C )
) )
103, 8, 9sylanbrc 662 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 367    C_ wss 3461   `'ccnv 4987    |` cres 4990   Fun wfun 5564   -->wf 5566   -1-1->wf1 5567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575
This theorem is referenced by:  f1ores  5812  oacomf1olem  7205  pwfseqlem5  9030  hashimarn  12480  hashf1lem2  12489  conjsubgen  16498  sylow1lem2  16818  sylow2blem1  16839  usgrares  24571  usgrares1  24612
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