Theorem fresf1o 29433

Description: Conditions for a restriction to be a one-to-one onto function. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Assertion

Ref	Expression
fresf1o	|- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ( F |` ( `' F " C ) ) : ( `' F " C ) -1-1-onto-> C )

Proof of Theorem fresf1o

Step	Hyp	Ref	Expression
1		funfn 5918	. . . . . . . 8 |- ( Fun ( `' F |` C ) <-> ( `' F |` C ) Fn dom ( `' F |` C ) )
2	1	biimpi 206	. . . . . . 7 |- ( Fun ( `' F |` C ) -> ( `' F |` C ) Fn dom ( `' F |` C ) )
3	2	3ad2ant3 1084	. . . . . 6 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ( `' F |` C ) Fn dom ( `' F |` C ) )
4		simp2 1062	. . . . . . . . 9 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> C C_ ran F )
5		df-rn 5125	. . . . . . . . 9 |- ran F = dom `' F
6	4, 5	syl6sseq 3651	. . . . . . . 8 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> C C_ dom `' F )
7		ssdmres 5420	. . . . . . . 8 |- ( C C_ dom `' F <-> dom ( `' F |` C ) = C )
8	6, 7	sylib 208	. . . . . . 7 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> dom ( `' F |` C ) = C )
9	8	fneq2d 5982	. . . . . 6 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ( ( `' F |` C ) Fn dom ( `' F |` C ) <-> ( `' F |` C ) Fn C ) )
10	3, 9	mpbid 222	. . . . 5 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ( `' F |` C ) Fn C )
11		simp1 1061	. . . . . . 7 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> Fun F )
12		funres 5929	. . . . . . 7 |- ( Fun F -> Fun ( F |` ( `' F " C ) ) )
13	11, 12	syl 17	. . . . . 6 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> Fun ( F |` ( `' F " C ) ) )
14		funcnvres2 5969	. . . . . . . 8 |- ( Fun F -> `' ( `' F |` C ) = ( F |` ( `' F " C ) ) )
15	11, 14	syl 17	. . . . . . 7 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> `' ( `' F |` C ) = ( F |` ( `' F " C ) ) )
16	15	funeqd 5910	. . . . . 6 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ( Fun `' ( `' F |` C ) <-> Fun ( F |` ( `' F " C ) ) ) )
17	13, 16	mpbird 247	. . . . 5 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> Fun `' ( `' F |` C ) )
18		df-ima 5127	. . . . . . 7 |- ( `' F " C ) = ran ( `' F |` C )
19	18	eqcomi 2631	. . . . . 6 |- ran ( `' F |` C ) = ( `' F " C )
20	19	a1i 11	. . . . 5 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ran ( `' F |` C ) = ( `' F " C ) )
21	10, 17, 20	3jca 1242	. . . 4 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ( ( `' F |` C ) Fn C /\ Fun `' ( `' F |` C ) /\ ran ( `' F |` C ) = ( `' F " C ) ) )
22		dff1o2 6142	. . . 4 |- ( ( `' F |` C ) : C -1-1-onto-> ( `' F " C ) <-> ( ( `' F |` C ) Fn C /\ Fun `' ( `' F |` C ) /\ ran ( `' F |` C ) = ( `' F " C ) ) )
23	21, 22	sylibr 224	. . 3 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ( `' F |` C ) : C -1-1-onto-> ( `' F " C ) )
24		f1ocnv 6149	. . 3 |- ( ( `' F |` C ) : C -1-1-onto-> ( `' F " C ) -> `' ( `' F |` C ) : ( `' F " C ) -1-1-onto-> C )
25	23, 24	syl 17	. 2 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> `' ( `' F |` C ) : ( `' F " C ) -1-1-onto-> C )
26		f1oeq1 6127	. . 3 |- ( `' ( `' F |` C ) = ( F |` ( `' F " C ) ) -> ( `' ( `' F |` C ) : ( `' F " C ) -1-1-onto-> C <-> ( F |` ( `' F " C ) ) : ( `' F " C ) -1-1-onto-> C ) )
27	11, 14, 26	3syl 18	. 2 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ( `' ( `' F |` C ) : ( `' F " C ) -1-1-onto-> C <-> ( F |` ( `' F " C ) ) : ( `' F " C ) -1-1-onto-> C ) )
28	25, 27	mpbid 222	1 |- ( ( Fun F /\ C C_ ran F /\ Fun ( `' F |` C ) ) -> ( F |` ( `' F " C ) ) : ( `' F " C ) -1-1-onto-> C )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   C_ wss 3574   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -1-1-onto->wf1o 5887

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-eu 2474  df-mo 2475  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  carsggect  30380