Theorem rinvf1o 6008

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 6007	. . . . 5 |- ( Fun F <-> F : dom F --> ran F )
3	1, 2	mpbi 145	. . . 4 |- F : dom F --> ran F
4		rinvbij.2	. . . . . 6 |- `' F = F
5	4	funeqi 5378	. . . . 5 |- ( Fun `' F <-> Fun F )
6	1, 5	mpbir 146	. . . 4 |- Fun `' F
7		df-f1 5362	. . . 4 |- ( F : dom F -1-1-> ran F <-> ( F : dom F --> ran F /\ Fun `' F ) )
8	3, 6, 7	mpbir2an 951	. . 3 |- F : dom F -1-1-> ran F
9		rinvbij.4a	. . 3 |- A C_ dom F
10		f1ores 5634	. . 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 426	. 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 5799	. . . . . . 7 |- ( ( Fun F /\ B C_ dom F ) -> ( ( F " B ) C_ A <-> B C_ ( `' F " A ) ) )
16	1, 14, 15	mp2an 426	. . . . . 6 |- ( ( F " B ) C_ A <-> B C_ ( `' F " A ) )
17	13, 16	mpbi 145	. . . . 5 |- B C_ ( `' F " A )
18	4	imaeq1i 5103	. . . . 5 |- ( `' F " A ) = ( F " A )
19	17, 18	sseqtri 3276	. . . 4 |- B C_ ( F " A )
20	12, 19	eqssi 3258	. . 3 |- ( F " A ) = B
21		f1oeq3 5609	. . 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 145	1 |- ( F |` A ) : A -1-1-onto-> B

Syntax hints:   <-> wb 105   = wceq 1398   C_ wss 3214   `'ccnv 4753   dom cdm 4754   ran crn 4755   |` cres 4756   "cima 4757   Fun wfun 5351   -->wf 5353   -1-1->wf1 5354   -1-1-onto->wf1o 5356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  ballotfilem7  13223