Theorem grpoidval 27367

Description: Lemma for grpoidcl 27368 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpoidval.1	|- X = ran G
grpoidval.2	|- U = (GId ` G )

Assertion

Ref	Expression
grpoidval	|- ( G e. GrpOp -> U = ( iota_ u e. X A. x e. X ( u G x ) = x ) )

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

Proof of Theorem grpoidval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpoidval.2	. 2 |- U = (GId ` G )
2		grpoidval.1	. . . 4 |- X = ran G
3	2	gidval 27366	. . 3 |- ( G e. GrpOp -> (GId ` G ) = ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
4		simpl 473	. . . . . . . . 9 |- ( ( ( u G x ) = x /\ ( x G u ) = x ) -> ( u G x ) = x )
5	4	ralimi 2952	. . . . . . . 8 |- ( A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> A. x e. X ( u G x ) = x )
6	5	rgenw 2924	. . . . . . 7 |- A. u e. X ( A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> A. x e. X ( u G x ) = x )
7	6	a1i 11	. . . . . 6 |- ( G e. GrpOp -> A. u e. X ( A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> A. x e. X ( u G x ) = x ) )
8	2	grpoidinv 27362	. . . . . . 7 |- ( G e. GrpOp -> E. u e. X A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) )
9		simpl 473	. . . . . . . . 9 |- ( ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) -> ( ( u G x ) = x /\ ( x G u ) = x ) )
10	9	ralimi 2952	. . . . . . . 8 |- ( A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) -> A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) )
11	10	reximi 3011	. . . . . . 7 |- ( E. u e. X A. x e. X ( ( ( u G x ) = x /\ ( x G u ) = x ) /\ E. y e. X ( ( y G x ) = u /\ ( x G y ) = u ) ) -> E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) )
12	8, 11	syl 17	. . . . . 6 |- ( G e. GrpOp -> E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) )
13	2	grpoideu 27363	. . . . . 6 |- ( G e. GrpOp -> E! u e. X A. x e. X ( u G x ) = x )
14	7, 12, 13	3jca 1242	. . . . 5 |- ( G e. GrpOp -> ( A. u e. X ( A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> A. x e. X ( u G x ) = x ) /\ E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) /\ E! u e. X A. x e. X ( u G x ) = x ) )
15		reupick2 3913	. . . . 5 |- ( ( ( A. u e. X ( A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> A. x e. X ( u G x ) = x ) /\ E. u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) /\ E! u e. X A. x e. X ( u G x ) = x ) /\ u e. X ) -> ( A. x e. X ( u G x ) = x <-> A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
16	14, 15	sylan 488	. . . 4 |- ( ( G e. GrpOp /\ u e. X ) -> ( A. x e. X ( u G x ) = x <-> A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
17	16	riotabidva 6627	. . 3 |- ( G e. GrpOp -> ( iota_ u e. X A. x e. X ( u G x ) = x ) = ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
18	3, 17	eqtr4d 2659	. 2 |- ( G e. GrpOp -> (GId ` G ) = ( iota_ u e. X A. x e. X ( u G x ) = x ) )
19	1, 18	syl5eq 2668	1 |- ( G e. GrpOp -> U = ( iota_ u e. X A. x e. X ( u G x ) = x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   ran crn 5115   ` cfv 5888   iota_crio 6610  (class class class)co 6650   GrpOpcgr 27343  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-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-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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-grpo 27347  df-gid 27348

This theorem is referenced by:  grpoidcl  27368  grpoidinv2  27369  cnidOLD  27437  hilid  28018