Theorem isexid 33646

Description: The predicate G has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
isexid.1	|- X = dom dom G

Assertion

Ref	Expression
isexid	|- ( G e. A -> ( G e. ExId <-> E. x e. X A. y e. X ( ( x G y ) = y /\ ( y G x ) = y ) ) )

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

Allowed substitution hints:   A( x, y)

Proof of Theorem isexid

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5324	. . . . 5 |- ( g = G -> dom g = dom G )
2	1	dmeqd 5326	. . . 4 |- ( g = G -> dom dom g = dom dom G )
3		isexid.1	. . . 4 |- X = dom dom G
4	2, 3	syl6eqr 2674	. . 3 |- ( g = G -> dom dom g = X )
5		oveq 6656	. . . . . 6 |- ( g = G -> ( x g y ) = ( x G y ) )
6	5	eqeq1d 2624	. . . . 5 |- ( g = G -> ( ( x g y ) = y <-> ( x G y ) = y ) )
7		oveq 6656	. . . . . 6 |- ( g = G -> ( y g x ) = ( y G x ) )
8	7	eqeq1d 2624	. . . . 5 |- ( g = G -> ( ( y g x ) = y <-> ( y G x ) = y ) )
9	6, 8	anbi12d 747	. . . 4 |- ( g = G -> ( ( ( x g y ) = y /\ ( y g x ) = y ) <-> ( ( x G y ) = y /\ ( y G x ) = y ) ) )
10	4, 9	raleqbidv 3152	. . 3 |- ( g = G -> ( A. y e. dom dom g ( ( x g y ) = y /\ ( y g x ) = y ) <-> A. y e. X ( ( x G y ) = y /\ ( y G x ) = y ) ) )
11	4, 10	rexeqbidv 3153	. 2 |- ( g = G -> ( E. x e. dom dom g A. y e. dom dom g ( ( x g y ) = y /\ ( y g x ) = y ) <-> E. x e. X A. y e. X ( ( x G y ) = y /\ ( y G x ) = y ) ) )
12		df-exid 33644	. 2 |- ExId = { g | E. x e. dom dom g A. y e. dom dom g ( ( x g y ) = y /\ ( y g x ) = y ) }
13	11, 12	elab2g 3353	1 |- ( G e. A -> ( G e. ExId <-> E. x e. X A. y e. X ( ( x G y ) = y /\ ( y G x ) = y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   dom cdm 5114  (class class class)co 6650   ExId cexid 33643

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-ral 2917  df-rex 2918  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-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-exid 33644

This theorem is referenced by:  opidonOLD  33651  isexid2  33654  ismndo  33671  exidres  33677