Theorem nvof1o 6536

Description: An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Assertion

Ref	Expression
nvof1o	|- ( ( F Fn A /\ `' F = F ) -> F : A -1-1-onto-> A )

Proof of Theorem nvof1o

Step	Hyp	Ref	Expression
1		fnfun 5988	. . . . . 6 |- ( F Fn A -> Fun F )
2		fdmrn 6064	. . . . . 6 |- ( Fun F <-> F : dom F --> ran F )
3	1, 2	sylib 208	. . . . 5 |- ( F Fn A -> F : dom F --> ran F )
4	3	adantr 481	. . . 4 |- ( ( F Fn A /\ `' F = F ) -> F : dom F --> ran F )
5		fndm 5990	. . . . . 6 |- ( F Fn A -> dom F = A )
6	5	adantr 481	. . . . 5 |- ( ( F Fn A /\ `' F = F ) -> dom F = A )
7		df-rn 5125	. . . . . . 7 |- ran F = dom `' F
8		dmeq 5324	. . . . . . 7 |- ( `' F = F -> dom `' F = dom F )
9	7, 8	syl5eq 2668	. . . . . 6 |- ( `' F = F -> ran F = dom F )
10	9, 5	sylan9eqr 2678	. . . . 5 |- ( ( F Fn A /\ `' F = F ) -> ran F = A )
11	6, 10	feq23d 6040	. . . 4 |- ( ( F Fn A /\ `' F = F ) -> ( F : dom F --> ran F <-> F : A --> A ) )
12	4, 11	mpbid 222	. . 3 |- ( ( F Fn A /\ `' F = F ) -> F : A --> A )
13	1	adantr 481	. . . 4 |- ( ( F Fn A /\ `' F = F ) -> Fun F )
14		funeq 5908	. . . . 5 |- ( `' F = F -> ( Fun `' F <-> Fun F ) )
15	14	adantl 482	. . . 4 |- ( ( F Fn A /\ `' F = F ) -> ( Fun `' F <-> Fun F ) )
16	13, 15	mpbird 247	. . 3 |- ( ( F Fn A /\ `' F = F ) -> Fun `' F )
17		df-f1 5893	. . 3 |- ( F : A -1-1-> A <-> ( F : A --> A /\ Fun `' F ) )
18	12, 16, 17	sylanbrc 698	. 2 |- ( ( F Fn A /\ `' F = F ) -> F : A -1-1-> A )
19		simpl 473	. . 3 |- ( ( F Fn A /\ `' F = F ) -> F Fn A )
20		df-fo 5894	. . 3 |- ( F : A -onto-> A <-> ( F Fn A /\ ran F = A ) )
21	19, 10, 20	sylanbrc 698	. 2 |- ( ( F Fn A /\ `' F = F ) -> F : A -onto-> A )
22		df-f1o 5895	. 2 |- ( F : A -1-1-onto-> A <-> ( F : A -1-1-> A /\ F : A -onto-> A ) )
23	18, 21, 22	sylanbrc 698	1 |- ( ( F Fn A /\ `' F = F ) -> F : A -1-1-onto-> A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   `'ccnv 5113   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -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

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-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-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  mirf1o  25564  lmif1o  25687  dssmapf1od  38315