Theorem grpoinvval 27377

Description: The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpinvfval.1	|- X = ran G
grpinvfval.2	|- U = (GId ` G )
grpinvfval.3	|- N = ( inv ` G )

Assertion

Ref	Expression
grpoinvval	|- ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) = ( iota_ y e. X ( y G A ) = U ) )

Distinct variable groups:   y, A   y, G   y, X

Allowed substitution hints:   U( y)   N( y)

Proof of Theorem grpoinvval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinvfval.1	. . . 4 |- X = ran G
2		grpinvfval.2	. . . 4 |- U = (GId ` G )
3		grpinvfval.3	. . . 4 |- N = ( inv ` G )
4	1, 2, 3	grpoinvfval 27376	. . 3 |- ( G e. GrpOp -> N = ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) ) )
5	4	fveq1d 6193	. 2 |- ( G e. GrpOp -> ( N ` A ) = ( ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) ) ` A ) )
6		oveq2 6658	. . . . 5 |- ( x = A -> ( y G x ) = ( y G A ) )
7	6	eqeq1d 2624	. . . 4 |- ( x = A -> ( ( y G x ) = U <-> ( y G A ) = U ) )
8	7	riotabidv 6613	. . 3 |- ( x = A -> ( iota_ y e. X ( y G x ) = U ) = ( iota_ y e. X ( y G A ) = U ) )
9		eqid 2622	. . 3 |- ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) ) = ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) )
10		riotaex 6615	. . 3 |- ( iota_ y e. X ( y G A ) = U ) e. _V
11	8, 9, 10	fvmpt 6282	. 2 |- ( A e. X -> ( ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) ) ` A ) = ( iota_ y e. X ( y G A ) = U ) )
12	5, 11	sylan9eq 2676	1 |- ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) = ( iota_ y e. X ( y G A ) = U ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ran crn 5115   ` 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-ginv 27349

This theorem is referenced by:  grpoinvcl  27378  grpoinv  27379