Theorem grpoinvf 27386

Description: Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpasscan1.1	|- X = ran G
grpasscan1.2	|- N = ( inv ` G )

Assertion

Ref	Expression
grpoinvf	|- ( G e. GrpOp -> N : X -1-1-onto-> X )

Proof of Theorem grpoinvf

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotaex 6615	. . . 4 |- ( iota_ y e. X ( y G x ) = (GId ` G ) ) e. _V
2		eqid 2622	. . . 4 |- ( x e. X |-> ( iota_ y e. X ( y G x ) = (GId ` G ) ) ) = ( x e. X |-> ( iota_ y e. X ( y G x ) = (GId ` G ) ) )
3	1, 2	fnmpti 6022	. . 3 |- ( x e. X |-> ( iota_ y e. X ( y G x ) = (GId ` G ) ) ) Fn X
4		grpasscan1.1	. . . . 5 |- X = ran G
5		eqid 2622	. . . . 5 |- (GId ` G ) = (GId ` G )
6		grpasscan1.2	. . . . 5 |- N = ( inv ` G )
7	4, 5, 6	grpoinvfval 27376	. . . 4 |- ( G e. GrpOp -> N = ( x e. X |-> ( iota_ y e. X ( y G x ) = (GId ` G ) ) ) )
8	7	fneq1d 5981	. . 3 |- ( G e. GrpOp -> ( N Fn X <-> ( x e. X |-> ( iota_ y e. X ( y G x ) = (GId ` G ) ) ) Fn X ) )
9	3, 8	mpbiri 248	. 2 |- ( G e. GrpOp -> N Fn X )
10		fnrnfv 6242	. . . 4 |- ( N Fn X -> ran N = { y | E. x e. X y = ( N ` x ) } )
11	9, 10	syl 17	. . 3 |- ( G e. GrpOp -> ran N = { y | E. x e. X y = ( N ` x ) } )
12	4, 6	grpoinvcl 27378	. . . . . . 7 |- ( ( G e. GrpOp /\ y e. X ) -> ( N ` y ) e. X )
13	4, 6	grpo2inv 27385	. . . . . . . 8 |- ( ( G e. GrpOp /\ y e. X ) -> ( N ` ( N ` y ) ) = y )
14	13	eqcomd 2628	. . . . . . 7 |- ( ( G e. GrpOp /\ y e. X ) -> y = ( N ` ( N ` y ) ) )
15		fveq2 6191	. . . . . . . . 9 |- ( x = ( N ` y ) -> ( N ` x ) = ( N ` ( N ` y ) ) )
16	15	eqeq2d 2632	. . . . . . . 8 |- ( x = ( N ` y ) -> ( y = ( N ` x ) <-> y = ( N ` ( N ` y ) ) ) )
17	16	rspcev 3309	. . . . . . 7 |- ( ( ( N ` y ) e. X /\ y = ( N ` ( N ` y ) ) ) -> E. x e. X y = ( N ` x ) )
18	12, 14, 17	syl2anc 693	. . . . . 6 |- ( ( G e. GrpOp /\ y e. X ) -> E. x e. X y = ( N ` x ) )
19	18	ex 450	. . . . 5 |- ( G e. GrpOp -> ( y e. X -> E. x e. X y = ( N ` x ) ) )
20		simpr 477	. . . . . . . 8 |- ( ( ( G e. GrpOp /\ x e. X ) /\ y = ( N ` x ) ) -> y = ( N ` x ) )
21	4, 6	grpoinvcl 27378	. . . . . . . . 9 |- ( ( G e. GrpOp /\ x e. X ) -> ( N ` x ) e. X )
22	21	adantr 481	. . . . . . . 8 |- ( ( ( G e. GrpOp /\ x e. X ) /\ y = ( N ` x ) ) -> ( N ` x ) e. X )
23	20, 22	eqeltrd 2701	. . . . . . 7 |- ( ( ( G e. GrpOp /\ x e. X ) /\ y = ( N ` x ) ) -> y e. X )
24	23	exp31 630	. . . . . 6 |- ( G e. GrpOp -> ( x e. X -> ( y = ( N ` x ) -> y e. X ) ) )
25	24	rexlimdv 3030	. . . . 5 |- ( G e. GrpOp -> ( E. x e. X y = ( N ` x ) -> y e. X ) )
26	19, 25	impbid 202	. . . 4 |- ( G e. GrpOp -> ( y e. X <-> E. x e. X y = ( N ` x ) ) )
27	26	abbi2dv 2742	. . 3 |- ( G e. GrpOp -> X = { y | E. x e. X y = ( N ` x ) } )
28	11, 27	eqtr4d 2659	. 2 |- ( G e. GrpOp -> ran N = X )
29		fveq2 6191	. . . 4 |- ( ( N ` x ) = ( N ` y ) -> ( N ` ( N ` x ) ) = ( N ` ( N ` y ) ) )
30	4, 6	grpo2inv 27385	. . . . . 6 |- ( ( G e. GrpOp /\ x e. X ) -> ( N ` ( N ` x ) ) = x )
31	30, 13	eqeqan12d 2638	. . . . 5 |- ( ( ( G e. GrpOp /\ x e. X ) /\ ( G e. GrpOp /\ y e. X ) ) -> ( ( N ` ( N ` x ) ) = ( N ` ( N ` y ) ) <-> x = y ) )
32	31	anandis 873	. . . 4 |- ( ( G e. GrpOp /\ ( x e. X /\ y e. X ) ) -> ( ( N ` ( N ` x ) ) = ( N ` ( N ` y ) ) <-> x = y ) )
33	29, 32	syl5ib 234	. . 3 |- ( ( G e. GrpOp /\ ( x e. X /\ y e. X ) ) -> ( ( N ` x ) = ( N ` y ) -> x = y ) )
34	33	ralrimivva 2971	. 2 |- ( G e. GrpOp -> A. x e. X A. y e. X ( ( N ` x ) = ( N ` y ) -> x = y ) )
35		dff1o6 6531	. 2 |- ( N : X -1-1-onto-> X <-> ( N Fn X /\ ran N = X /\ A. x e. X A. y e. X ( ( N ` x ) = ( N ` y ) -> x = y ) ) )
36	9, 28, 34, 35	syl3anbrc 1246	1 |- ( G e. GrpOp -> N : X -1-1-onto-> X )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888   iota_crio 6610  (class class class)co 6650   GrpOpcgr 27343  GIdcgi 27344   invcgn 27345

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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  df-riota 6611  df-ov 6653  df-grpo 27347  df-gid 27348  df-ginv 27349

This theorem is referenced by:  nvinvfval  27495