Theorem exidreslem 33676

Description: Lemma for exidres 33677 and exidresid 33678. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)

Hypotheses

Ref	Expression
exidres.1	|- X = ran G
exidres.2	|- U = (GId ` G )
exidres.3	|- H = ( G |` ( Y X. Y ) )

Assertion

Ref	Expression
exidreslem	|- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> ( U e. dom dom H /\ A. x e. dom dom H ( ( U H x ) = x /\ ( x H U ) = x ) ) )

Distinct variable groups:   x, G   x, Y   x, X   x, U   x, H

Proof of Theorem exidreslem

Step	Hyp	Ref	Expression
1		exidres.3	. . . . . . . 8 |- H = ( G |` ( Y X. Y ) )
2	1	dmeqi 5325	. . . . . . 7 |- dom H = dom ( G |` ( Y X. Y ) )
3		xpss12 5225	. . . . . . . . . . 11 |- ( ( Y C_ X /\ Y C_ X ) -> ( Y X. Y ) C_ ( X X. X ) )
4	3	anidms 677	. . . . . . . . . 10 |- ( Y C_ X -> ( Y X. Y ) C_ ( X X. X ) )
5		exidres.1	. . . . . . . . . . . . 13 |- X = ran G
6	5	opidon2OLD 33653	. . . . . . . . . . . 12 |- ( G e. ( Magma i^i ExId ) -> G : ( X X. X ) -onto-> X )
7		fof 6115	. . . . . . . . . . . 12 |- ( G : ( X X. X ) -onto-> X -> G : ( X X. X ) --> X )
8		fdm 6051	. . . . . . . . . . . 12 |- ( G : ( X X. X ) --> X -> dom G = ( X X. X ) )
9	6, 7, 8	3syl 18	. . . . . . . . . . 11 |- ( G e. ( Magma i^i ExId ) -> dom G = ( X X. X ) )
10	9	sseq2d 3633	. . . . . . . . . 10 |- ( G e. ( Magma i^i ExId ) -> ( ( Y X. Y ) C_ dom G <-> ( Y X. Y ) C_ ( X X. X ) ) )
11	4, 10	syl5ibr 236	. . . . . . . . 9 |- ( G e. ( Magma i^i ExId ) -> ( Y C_ X -> ( Y X. Y ) C_ dom G ) )
12	11	imp 445	. . . . . . . 8 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) -> ( Y X. Y ) C_ dom G )
13		ssdmres 5420	. . . . . . . 8 |- ( ( Y X. Y ) C_ dom G <-> dom ( G |` ( Y X. Y ) ) = ( Y X. Y ) )
14	12, 13	sylib 208	. . . . . . 7 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) -> dom ( G |` ( Y X. Y ) ) = ( Y X. Y ) )
15	2, 14	syl5eq 2668	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) -> dom H = ( Y X. Y ) )
16	15	dmeqd 5326	. . . . 5 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) -> dom dom H = dom ( Y X. Y ) )
17		dmxpid 5345	. . . . 5 |- dom ( Y X. Y ) = Y
18	16, 17	syl6eq 2672	. . . 4 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) -> dom dom H = Y )
19	18	eleq2d 2687	. . 3 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) -> ( U e. dom dom H <-> U e. Y ) )
20	19	biimp3ar 1433	. 2 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> U e. dom dom H )
21		ssel2 3598	. . . . . . . . . 10 |- ( ( Y C_ X /\ x e. Y ) -> x e. X )
22		exidres.2	. . . . . . . . . . 11 |- U = (GId ` G )
23	5, 22	cmpidelt 33658	. . . . . . . . . 10 |- ( ( G e. ( Magma i^i ExId ) /\ x e. X ) -> ( ( U G x ) = x /\ ( x G U ) = x ) )
24	21, 23	sylan2 491	. . . . . . . . 9 |- ( ( G e. ( Magma i^i ExId ) /\ ( Y C_ X /\ x e. Y ) ) -> ( ( U G x ) = x /\ ( x G U ) = x ) )
25	24	anassrs 680	. . . . . . . 8 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) /\ x e. Y ) -> ( ( U G x ) = x /\ ( x G U ) = x ) )
26	25	adantrl 752	. . . . . . 7 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) /\ ( U e. Y /\ x e. Y ) ) -> ( ( U G x ) = x /\ ( x G U ) = x ) )
27	1	oveqi 6663	. . . . . . . . . . 11 |- ( U H x ) = ( U ( G |` ( Y X. Y ) ) x )
28		ovres 6800	. . . . . . . . . . 11 |- ( ( U e. Y /\ x e. Y ) -> ( U ( G |` ( Y X. Y ) ) x ) = ( U G x ) )
29	27, 28	syl5eq 2668	. . . . . . . . . 10 |- ( ( U e. Y /\ x e. Y ) -> ( U H x ) = ( U G x ) )
30	29	eqeq1d 2624	. . . . . . . . 9 |- ( ( U e. Y /\ x e. Y ) -> ( ( U H x ) = x <-> ( U G x ) = x ) )
31	1	oveqi 6663	. . . . . . . . . . . 12 |- ( x H U ) = ( x ( G |` ( Y X. Y ) ) U )
32		ovres 6800	. . . . . . . . . . . 12 |- ( ( x e. Y /\ U e. Y ) -> ( x ( G |` ( Y X. Y ) ) U ) = ( x G U ) )
33	31, 32	syl5eq 2668	. . . . . . . . . . 11 |- ( ( x e. Y /\ U e. Y ) -> ( x H U ) = ( x G U ) )
34	33	ancoms 469	. . . . . . . . . 10 |- ( ( U e. Y /\ x e. Y ) -> ( x H U ) = ( x G U ) )
35	34	eqeq1d 2624	. . . . . . . . 9 |- ( ( U e. Y /\ x e. Y ) -> ( ( x H U ) = x <-> ( x G U ) = x ) )
36	30, 35	anbi12d 747	. . . . . . . 8 |- ( ( U e. Y /\ x e. Y ) -> ( ( ( U H x ) = x /\ ( x H U ) = x ) <-> ( ( U G x ) = x /\ ( x G U ) = x ) ) )
37	36	adantl 482	. . . . . . 7 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) /\ ( U e. Y /\ x e. Y ) ) -> ( ( ( U H x ) = x /\ ( x H U ) = x ) <-> ( ( U G x ) = x /\ ( x G U ) = x ) ) )
38	26, 37	mpbird 247	. . . . . 6 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) /\ ( U e. Y /\ x e. Y ) ) -> ( ( U H x ) = x /\ ( x H U ) = x ) )
39	38	anassrs 680	. . . . 5 |- ( ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) /\ U e. Y ) /\ x e. Y ) -> ( ( U H x ) = x /\ ( x H U ) = x ) )
40	39	ralrimiva 2966	. . . 4 |- ( ( ( G e. ( Magma i^i ExId ) /\ Y C_ X ) /\ U e. Y ) -> A. x e. Y ( ( U H x ) = x /\ ( x H U ) = x ) )
41	40	3impa 1259	. . 3 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> A. x e. Y ( ( U H x ) = x /\ ( x H U ) = x ) )
42	12	3adant3 1081	. . . . . . . 8 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> ( Y X. Y ) C_ dom G )
43	42, 13	sylib 208	. . . . . . 7 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> dom ( G |` ( Y X. Y ) ) = ( Y X. Y ) )
44	2, 43	syl5eq 2668	. . . . . 6 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> dom H = ( Y X. Y ) )
45	44	dmeqd 5326	. . . . 5 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> dom dom H = dom ( Y X. Y ) )
46	45, 17	syl6eq 2672	. . . 4 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> dom dom H = Y )
47	46	raleqdv 3144	. . 3 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> ( A. x e. dom dom H ( ( U H x ) = x /\ ( x H U ) = x ) <-> A. x e. Y ( ( U H x ) = x /\ ( x H U ) = x ) ) )
48	41, 47	mpbird 247	. 2 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> A. x e. dom dom H ( ( U H x ) = x /\ ( x H U ) = x ) )
49	20, 48	jca 554	1 |- ( ( G e. ( Magma i^i ExId ) /\ Y C_ X /\ U e. Y ) -> ( U e. dom dom H /\ A. x e. dom dom H ( ( U H x ) = x /\ ( x H U ) = x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650  GIdcgi 27344   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-reu 2919  df-rmo 2920  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-gid 27348  df-exid 33644  df-mgmOLD 33648

This theorem is referenced by:  exidres  33677  exidresid  33678