Theorem gidval 27366

Description: The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
gidval.1	|- X = ran G

Assertion

Ref	Expression
gidval	|- ( G e. V -> (GId ` G ) = ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )

Distinct variable groups:   x, u, G   u, X, x

Allowed substitution hints:   V( x, u)

Proof of Theorem gidval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( G e. V -> G e. _V )
2		rneq 5351	. . . . 5 |- ( g = G -> ran g = ran G )
3		gidval.1	. . . . 5 |- X = ran G
4	2, 3	syl6eqr 2674	. . . 4 |- ( g = G -> ran g = X )
5		oveq 6656	. . . . . . 7 |- ( g = G -> ( u g x ) = ( u G x ) )
6	5	eqeq1d 2624	. . . . . 6 |- ( g = G -> ( ( u g x ) = x <-> ( u G x ) = x ) )
7		oveq 6656	. . . . . . 7 |- ( g = G -> ( x g u ) = ( x G u ) )
8	7	eqeq1d 2624	. . . . . 6 |- ( g = G -> ( ( x g u ) = x <-> ( x G u ) = x ) )
9	6, 8	anbi12d 747	. . . . 5 |- ( g = G -> ( ( ( u g x ) = x /\ ( x g u ) = x ) <-> ( ( u G x ) = x /\ ( x G u ) = x ) ) )
10	4, 9	raleqbidv 3152	. . . 4 |- ( g = G -> ( A. x e. ran g ( ( u g x ) = x /\ ( x g u ) = x ) <-> A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
11	4, 10	riotaeqbidv 6614	. . 3 |- ( g = G -> ( iota_ u e. ran g A. x e. ran g ( ( u g x ) = x /\ ( x g u ) = x ) ) = ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
12		df-gid 27348	. . 3 |- GId = ( g e. _V |-> ( iota_ u e. ran g A. x e. ran g ( ( u g x ) = x /\ ( x g u ) = x ) ) )
13		riotaex 6615	. . 3 |- ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) e. _V
14	11, 12, 13	fvmpt 6282	. 2 |- ( G e. _V -> (GId ` G ) = ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
15	1, 14	syl 17	1 |- ( G e. V -> (GId ` G ) = ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   ran crn 5115   ` cfv 5888   iota_crio 6610  (class class class)co 6650  GIdcgi 27344

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-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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-gid 27348

This theorem is referenced by:  grpoidval  27367  idrval  33656  exidresid  33678