Theorem isexid2 33654

Description: If G e. ( Magma i^i ExId ), then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
isexid2.1	|- X = ran G

Assertion

Ref	Expression
isexid2	|- ( G e. ( Magma i^i ExId ) -> E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) )

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

Proof of Theorem isexid2

Step	Hyp	Ref	Expression
1		isexid2.1	. 2 |- X = ran G
2		rngopidOLD 33652	. . . . 5 |- ( G e. ( Magma i^i ExId ) -> ran G = dom dom G )
3		elin 3796	. . . . . . 7 |- ( G e. ( Magma i^i ExId ) <-> ( G e. Magma /\ G e. ExId ) )
4		eqid 2622	. . . . . . . . . . 11 |- dom dom G = dom dom G
5	4	isexid 33646	. . . . . . . . . 10 |- ( G e. ExId -> ( G e. ExId <-> E. u e. dom dom G A. x e. dom dom G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
6	5	ibi 256	. . . . . . . . 9 |- ( G e. ExId -> E. u e. dom dom G A. x e. dom dom G ( ( u G x ) = x /\ ( x G u ) = x ) )
7	6	a1d 25	. . . . . . . 8 |- ( G e. ExId -> ( X = dom dom G -> E. u e. dom dom G A. x e. dom dom G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
8	7	adantl 482	. . . . . . 7 |- ( ( G e. Magma /\ G e. ExId ) -> ( X = dom dom G -> E. u e. dom dom G A. x e. dom dom G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
9	3, 8	sylbi 207	. . . . . 6 |- ( G e. ( Magma i^i ExId ) -> ( X = dom dom G -> E. u e. dom dom G A. x e. dom dom G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
10		eqeq2 2633	. . . . . . 7 |- ( ran G = dom dom G -> ( X = ran G <-> X = dom dom G ) )
11		raleq 3138	. . . . . . . 8 |- ( ran G = dom dom G -> ( A. x e. ran G ( ( u G x ) = x /\ ( x G u ) = x ) <-> A. x e. dom dom G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
12	11	rexeqbi1dv 3147	. . . . . . 7 |- ( ran G = dom dom G -> ( E. u e. ran G A. x e. ran G ( ( u G x ) = x /\ ( x G u ) = x ) <-> E. u e. dom dom G A. x e. dom dom G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
13	10, 12	imbi12d 334	. . . . . 6 |- ( ran G = dom dom G -> ( ( X = ran G -> E. u e. ran G A. x e. ran G ( ( u G x ) = x /\ ( x G u ) = x ) ) <-> ( X = dom dom G -> E. u e. dom dom G A. x e. dom dom G ( ( u G x ) = x /\ ( x G u ) = x ) ) ) )
14	9, 13	syl5ibr 236	. . . . 5 |- ( ran G = dom dom G -> ( G e. ( Magma i^i ExId ) -> ( X = ran G -> E. u e. ran G A. x e. ran G ( ( u G x ) = x /\ ( x G u ) = x ) ) ) )
15	2, 14	mpcom 38	. . . 4 |- ( G e. ( Magma i^i ExId ) -> ( X = ran G -> E. u e. ran G A. x e. ran G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
16	15	com12 32	. . 3 |- ( X = ran G -> ( G e. ( Magma i^i ExId ) -> E. u e. ran G A. x e. ran G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
17		raleq 3138	. . . 4 |- ( X = ran G -> ( A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) <-> A. x e. ran G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
18	17	rexeqbi1dv 3147	. . 3 |- ( X = ran G -> ( E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) <-> E. u e. ran G A. x e. ran G ( ( u G x ) = x /\ ( x G u ) = x ) ) )
19	16, 18	sylibrd 249	. 2 |- ( X = ran G -> ( G e. ( Magma i^i ExId ) -> E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
20	1, 19	ax-mp 5	1 |- ( G e. ( Magma i^i ExId ) -> E. 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   E.wrex 2913   i^i cin 3573   dom cdm 5114   ran crn 5115  (class class class)co 6650   ExId cexid 33643   Magmacmagm 33647

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-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-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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-exid 33644  df-mgmOLD 33648

This theorem is referenced by:  exidu1  33655