Theorem symgfo 14730

Description: The operation of a symetry group is onto.

Hypotheses

Ref	Expression
symgfo.1	|- A e. _V
symgfo.2	|- P = {x | x:A-1-1-onto->A}

Assertion

Ref	Expression
symgfo	|- (SymGrp` A):(P X. P)-onto->P

Distinct variable group:   x,A

Proof of Theorem symgfo

Step	Hyp	Ref	Expression
1		elin 2786	. . 3 |- ((SymGrp` A) e. (Magma i^i ExId ) <-> ((SymGrp` A) e. Magma /\ (SymGrp` A) e. ExId ))
2		symgfo.1	. . . . 5 |- A e. _V
3		symgfo.2	. . . . 5 |- P = {x | x:A-1-1-onto->A}
4	2, 3	symgf 10204	. . . 4 |- (SymGrp` A):(P X. P)-->P
5	2	symggrpi 10205	. . . . 5 |- (SymGrp` A) e. Grp
6		fdm 4567	. . . . . . . 8 |- ((SymGrp` A):(P X. P)-->P -> dom (SymGrp` A) = (P X. P))
7		dmeq 4157	. . . . . . . 8 |- (dom (SymGrp` A) = (P X. P) -> dom dom (SymGrp` A) = dom ( P X. P))
8		dmxpid 4179	. . . . . . . . 9 |- dom ( P X. P) = P
9		eqtr 1904	. . . . . . . . . . . . 13 |- ((P = dom ( P X. P) /\ dom ( P X. P) = dom dom (SymGrp` A)) -> P = dom dom (SymGrp` A))
10	9	expcom 403	. . . . . . . . . . . 12 |- (dom ( P X. P) = dom dom (SymGrp` A) -> (P = dom ( P X. P) -> P = dom dom (SymGrp` A)))
11	10	eqcoms 1887	. . . . . . . . . . 11 |- (dom dom (SymGrp` A) = dom ( P X. P) -> (P = dom ( P X. P) -> P = dom dom (SymGrp` A)))
12	11	com12 14	. . . . . . . . . 10 |- (P = dom ( P X. P) -> (dom dom (SymGrp` A) = dom ( P X. P) -> P = dom dom (SymGrp` A)))
13	12	eqcoms 1887	. . . . . . . . 9 |- (dom ( P X. P) = P -> (dom dom (SymGrp` A) = dom ( P X. P) -> P = dom dom (SymGrp` A)))
14	8, 13	ax-mp 7	. . . . . . . 8 |- (dom dom (SymGrp` A) = dom ( P X. P) -> P = dom dom (SymGrp` A))
15	6, 7, 14	3syl 24	. . . . . . 7 |- ((SymGrp` A):(P X. P)-->P -> P = dom dom (SymGrp` A))
16	4, 15	ax-mp 7	. . . . . 6 |- P = dom dom (SymGrp` A)
17	16	ismgm 10367	. . . . 5 |- ((SymGrp` A) e. Grp -> ((SymGrp` A) e. Magma <-> (SymGrp` A):(P X. P)-->P))
18	5, 17	ax-mp 7	. . . 4 |- ((SymGrp` A) e. Magma <-> (SymGrp` A):(P X. P)-->P)
19	4, 18	mpbir 207	. . 3 |- (SymGrp` A) e. Magma
20		f1oi 4671	. . . . . . 7 |- ( _I |` A):A-1-1-onto->A
21		eqid 1884	. . . . . . . 8 |- {x | x:A-1-1-onto->A} = {x | x:A-1-1-onto->A}
22	2, 21	elsymgrn 10200	. . . . . . 7 |- (( _I |` A) e. {x | x:A-1-1-onto->A} <-> ( _I |` A):A-1-1-onto->A)
23	20, 22	mpbir 207	. . . . . 6 |- ( _I |` A) e. {x | x:A-1-1-onto->A}
24	23, 3	eleqtrri 1970	. . . . 5 |- ( _I |` A) e. P
25	2, 3	symgoprv 10203	. . . . . . . . 9 |- ((g e. P /\ ( _I |` A) e. P) -> (g(SymGrp` A)( _I |` A)) = (g o. ( _I |` A)))
26	3	eleq2i 1961	. . . . . . . . . . 11 |- (g e. P <-> g e. {x | x:A-1-1-onto->A})
27	2, 21	elsymgrn 10200	. . . . . . . . . . . 12 |- (g e. {x | x:A-1-1-onto->A} <-> g:A-1-1-onto->A)
28		f1of 4635	. . . . . . . . . . . . 13 |- (g:A-1-1-onto->A -> g:A-->A)
29		fcoi1 4584	. . . . . . . . . . . . 13 |- (g:A-->A -> (g o. ( _I |` A)) = g)
30	28, 29	syl 12	. . . . . . . . . . . 12 |- (g:A-1-1-onto->A -> (g o. ( _I |` A)) = g)
31	27, 30	sylbi 216	. . . . . . . . . . 11 |- (g e. {x | x:A-1-1-onto->A} -> (g o. ( _I |` A)) = g)
32	26, 31	sylbi 216	. . . . . . . . . 10 |- (g e. P -> (g o. ( _I |` A)) = g)
33	32	adantr 425	. . . . . . . . 9 |- ((g e. P /\ ( _I |` A) e. P) -> (g o. ( _I |` A)) = g)
34	25, 33	eqtrd 1925	. . . . . . . 8 |- ((g e. P /\ ( _I |` A) e. P) -> (g(SymGrp` A)( _I |` A)) = g)
35	24, 34	mpan2 760	. . . . . . 7 |- (g e. P -> (g(SymGrp` A)( _I |` A)) = g)
36	2, 3	symgoprv 10203	. . . . . . . . 9 |- ((( _I |` A) e. P /\ g e. P) -> (( _I |` A)(SymGrp` A)g) = (( _I |` A) o. g))
37		fcoi2 4586	. . . . . . . . . . . . 13 |- (g:A-->A -> (( _I |` A) o. g) = g)
38	28, 37	syl 12	. . . . . . . . . . . 12 |- (g:A-1-1-onto->A -> (( _I |` A) o. g) = g)
39	27, 38	sylbi 216	. . . . . . . . . . 11 |- (g e. {x | x:A-1-1-onto->A} -> (( _I |` A) o. g) = g)
40	26, 39	sylbi 216	. . . . . . . . . 10 |- (g e. P -> (( _I |` A) o. g) = g)
41	40	adantl 424	. . . . . . . . 9 |- ((( _I |` A) e. P /\ g e. P) -> (( _I |` A) o. g) = g)
42	36, 41	eqtrd 1925	. . . . . . . 8 |- ((( _I |` A) e. P /\ g e. P) -> (( _I |` A)(SymGrp` A)g) = g)
43	24, 42	mpan 759	. . . . . . 7 |- (g e. P -> (( _I |` A)(SymGrp` A)g) = g)
44	35, 43	jca 310	. . . . . 6 |- (g e. P -> ((g(SymGrp` A)( _I |` A)) = g /\ (( _I |` A)(SymGrp` A)g) = g))
45	44	rgen 2159	. . . . 5 |- A.g e. P ((g(SymGrp` A)( _I |` A)) = g /\ (( _I |` A)(SymGrp` A)g) = g)
46		opreq2 4890	. . . . . . . . 9 |- (f = ( _I |` A) -> (g(SymGrp` A)f) = (g(SymGrp` A)( _I |` A)))
47	46	eqeq1d 1892	. . . . . . . 8 |- (f = ( _I |` A) -> ((g(SymGrp` A)f) = g <-> (g(SymGrp` A)( _I |` A)) = g))
48		opreq1 4889	. . . . . . . . 9 |- (f = ( _I |` A) -> (f(SymGrp` A)g) = (( _I |` A)(SymGrp` A)g))
49	48	eqeq1d 1892	. . . . . . . 8 |- (f = ( _I |` A) -> ((f(SymGrp` A)g) = g <-> (( _I |` A)(SymGrp` A)g) = g))
50	47, 49	anbi12d 690	. . . . . . 7 |- (f = ( _I |` A) -> (((g(SymGrp` A)f) = g /\ (f(SymGrp` A)g) = g) <-> ((g(SymGrp` A)( _I |` A)) = g /\ (( _I |` A)(SymGrp` A)g) = g)))
51	50	ralbidv 2123	. . . . . 6 |- (f = ( _I |` A) -> (A.g e. P ((g(SymGrp` A)f) = g /\ (f(SymGrp` A)g) = g) <-> A.g e. P ((g(SymGrp` A)( _I |` A)) = g /\ (( _I |` A)(SymGrp` A)g) = g)))
52	51	rcla4ev 2381	. . . . 5 |- ((( _I |` A) e. P /\ A.g e. P ((g(SymGrp` A)( _I |` A)) = g /\ (( _I |` A)(SymGrp` A)g) = g)) -> E.f e. P A.g e. P ((g(SymGrp` A)f) = g /\ (f(SymGrp` A)g) = g))
53	24, 45, 52	mp2an 761	. . . 4 |- E.f e. P A.g e. P ((g(SymGrp` A)f) = g /\ (f(SymGrp` A)g) = g)
54	16	isexid 10364	. . . . 5 |- ((SymGrp` A) e. Grp -> ((SymGrp` A) e. ExId <-> E.f e. P A.g e. P ((g(SymGrp` A)f) = g /\ (f(SymGrp` A)g) = g)))
55	5, 54	ax-mp 7	. . . 4 |- ((SymGrp` A) e. ExId <-> E.f e. P A.g e. P ((g(SymGrp` A)f) = g /\ (f(SymGrp` A)g) = g))
56	53, 55	mpbir 207	. . 3 |- (SymGrp` A) e. ExId
57	1, 19, 56	mpbir2an 800	. 2 |- (SymGrp` A) e. (Magma i^i ExId )
58	16	opidon 10369	. 2 |- ((SymGrp` A) e. (Magma i^i ExId ) -> (SymGrp` A):(P X. P)-onto->P)
59	57, 58	ax-mp 7	1 |- (SymGrp` A):(P X. P)-onto->P

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592   _I cid 3582   X. cxp 3984  dom cdm 3986   |` cres 3988   o. ccom 3990  -->wf 3994  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  SymGrpcsymgrp 10198   ExId cexid 10361  Magmacmagm 10365

This theorem is referenced by:  curgrpact 14735

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-grp 9316  df-symgrp 10199  df-exid 10362  df-mgm 10366

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