Theorem f1ssres 6108

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

Step	Hyp	Ref	Expression
1		f1f 6101	. . 3 |- ( F : A -1-1-> B -> F : A --> B )
2		fssres 6070	. . 3 |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )
3	1, 2	sylan 488	. 2 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C --> B )
4		df-f1 5893	. . . . 5 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5	4	simprbi 480	. . . 4 |- ( F : A -1-1-> B -> Fun `' F )
6		funres11 5966	. . . 4 |- ( Fun `' F -> Fun `' ( F |` C ) )
7	5, 6	syl 17	. . 3 |- ( F : A -1-1-> B -> Fun `' ( F |` C ) )
8	7	adantr 481	. 2 |- ( ( F : A -1-1-> B /\ C C_ A ) -> Fun `' ( F |` C ) )
9		df-f1 5893	. 2 |- ( ( F |` C ) : C -1-1-> B <-> ( ( F |` C ) : C --> B /\ Fun `' ( F |` C ) ) )
10	3, 8, 9	sylanbrc 698	1 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-> B )

Syntax hints:   -> wi 4   /\ wa 384   C_ wss 3574   `'ccnv 5113   |` cres 5116   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893

This theorem is referenced by:  f1ores  6151  oacomf1olem  7644  pwfseqlem5  9485  hashimarn  13227  hashf1lem2  13240  conjsubgen  17693  sylow1lem2  18014  sylow2blem1  18035  usgrres  26200