Theorem rinvf1o 29432

Description: Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Hypotheses

Ref	Expression
rinvbij.1	|- Fun F
rinvbij.2	|- `' F = F
rinvbij.3a	|- ( F " A ) C_ B
rinvbij.3b	|- ( F " B ) C_ A
rinvbij.4a	|- A C_ dom F
rinvbij.4b	|- B C_ dom F

Assertion

Ref	Expression
rinvf1o	|- ( F |` A ) : A -1-1-onto-> B

Proof of Theorem rinvf1o

Step	Hyp	Ref	Expression
1		rinvbij.1	. . . . 5 |- Fun F
2		fdmrn 6064	. . . . 5 |- ( Fun F <-> F : dom F --> ran F )
3	1, 2	mpbi 220	. . . 4 |- F : dom F --> ran F
4		rinvbij.2	. . . . . 6 |- `' F = F
5	4	funeqi 5909	. . . . 5 |- ( Fun `' F <-> Fun F )
6	1, 5	mpbir 221	. . . 4 |- Fun `' F
7		df-f1 5893	. . . 4 |- ( F : dom F -1-1-> ran F <-> ( F : dom F --> ran F /\ Fun `' F ) )
8	3, 6, 7	mpbir2an 955	. . 3 |- F : dom F -1-1-> ran F
9		rinvbij.4a	. . 3 |- A C_ dom F
10		f1ores 6151	. . 3 |- ( ( F : dom F -1-1-> ran F /\ A C_ dom F ) -> ( F |` A ) : A -1-1-onto-> ( F " A ) )
11	8, 9, 10	mp2an 708	. 2 |- ( F |` A ) : A -1-1-onto-> ( F " A )
12		rinvbij.3a	. . . 4 |- ( F " A ) C_ B
13		rinvbij.3b	. . . . . 6 |- ( F " B ) C_ A
14		rinvbij.4b	. . . . . . 7 |- B C_ dom F
15		funimass3 6333	. . . . . . 7 |- ( ( Fun F /\ B C_ dom F ) -> ( ( F " B ) C_ A <-> B C_ ( `' F " A ) ) )
16	1, 14, 15	mp2an 708	. . . . . 6 |- ( ( F " B ) C_ A <-> B C_ ( `' F " A ) )
17	13, 16	mpbi 220	. . . . 5 |- B C_ ( `' F " A )
18	4	imaeq1i 5463	. . . . 5 |- ( `' F " A ) = ( F " A )
19	17, 18	sseqtri 3637	. . . 4 |- B C_ ( F " A )
20	12, 19	eqssi 3619	. . 3 |- ( F " A ) = B
21		f1oeq3 6129	. . 3 |- ( ( F " A ) = B -> ( ( F |` A ) : A -1-1-onto-> ( F " A ) <-> ( F |` A ) : A -1-1-onto-> B ) )
22	20, 21	ax-mp 5	. 2 |- ( ( F |` A ) : A -1-1-onto-> ( F " A ) <-> ( F |` A ) : A -1-1-onto-> B )
23	11, 22	mpbi 220	1 |- ( F |` A ) : A -1-1-onto-> B

Syntax hints:   <-> wb 196   = wceq 1483   C_ wss 3574   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  ballotlem7  30597