Theorem ismgmOLD 33649

Description: Obsolete version of ismgm 17243 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
ismgmOLD	|- ( G e. A -> ( G e. Magma <-> G : ( X X. X ) --> X ) )

Proof of Theorem ismgmOLD

Dummy variables g t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		feq1 6026	. . . . 5 |- ( g = G -> ( g : ( t X. t ) --> t <-> G : ( t X. t ) --> t ) )
2	1	exbidv 1850	. . . 4 |- ( g = G -> ( E. t g : ( t X. t ) --> t <-> E. t G : ( t X. t ) --> t ) )
3		df-mgmOLD 33648	. . . 4 |- Magma = { g | E. t g : ( t X. t ) --> t }
4	2, 3	elab2g 3353	. . 3 |- ( G e. A -> ( G e. Magma <-> E. t G : ( t X. t ) --> t ) )
5		f00 6087	. . . . . . . 8 |- ( G : ( (/) X. (/) ) --> (/) <-> ( G = (/) /\ ( (/) X. (/) ) = (/) ) )
6		dmeq 5324	. . . . . . . . . 10 |- ( G = (/) -> dom G = dom (/) )
7		dmeq 5324	. . . . . . . . . . 11 |- ( dom G = dom (/) -> dom dom G = dom dom (/) )
8		dm0 5339	. . . . . . . . . . . . 13 |- dom (/) = (/)
9	8	dmeqi 5325	. . . . . . . . . . . 12 |- dom dom (/) = dom (/)
10	9, 8	eqtri 2644	. . . . . . . . . . 11 |- dom dom (/) = (/)
11	7, 10	syl6req 2673	. . . . . . . . . 10 |- ( dom G = dom (/) -> (/) = dom dom G )
12	6, 11	syl 17	. . . . . . . . 9 |- ( G = (/) -> (/) = dom dom G )
13	12	adantr 481	. . . . . . . 8 |- ( ( G = (/) /\ ( (/) X. (/) ) = (/) ) -> (/) = dom dom G )
14	5, 13	sylbi 207	. . . . . . 7 |- ( G : ( (/) X. (/) ) --> (/) -> (/) = dom dom G )
15		xpeq12 5134	. . . . . . . . . 10 |- ( ( t = (/) /\ t = (/) ) -> ( t X. t ) = ( (/) X. (/) ) )
16	15	anidms 677	. . . . . . . . 9 |- ( t = (/) -> ( t X. t ) = ( (/) X. (/) ) )
17		feq23 6029	. . . . . . . . 9 |- ( ( ( t X. t ) = ( (/) X. (/) ) /\ t = (/) ) -> ( G : ( t X. t ) --> t <-> G : ( (/) X. (/) ) --> (/) ) )
18	16, 17	mpancom 703	. . . . . . . 8 |- ( t = (/) -> ( G : ( t X. t ) --> t <-> G : ( (/) X. (/) ) --> (/) ) )
19		eqeq1 2626	. . . . . . . 8 |- ( t = (/) -> ( t = dom dom G <-> (/) = dom dom G ) )
20	18, 19	imbi12d 334	. . . . . . 7 |- ( t = (/) -> ( ( G : ( t X. t ) --> t -> t = dom dom G ) <-> ( G : ( (/) X. (/) ) --> (/) -> (/) = dom dom G ) ) )
21	14, 20	mpbiri 248	. . . . . 6 |- ( t = (/) -> ( G : ( t X. t ) --> t -> t = dom dom G ) )
22		fdm 6051	. . . . . . . 8 |- ( G : ( t X. t ) --> t -> dom G = ( t X. t ) )
23		dmeq 5324	. . . . . . . 8 |- ( dom G = ( t X. t ) -> dom dom G = dom ( t X. t ) )
24		df-ne 2795	. . . . . . . . . . . 12 |- ( t =/= (/) <-> -. t = (/) )
25		dmxp 5344	. . . . . . . . . . . 12 |- ( t =/= (/) -> dom ( t X. t ) = t )
26	24, 25	sylbir 225	. . . . . . . . . . 11 |- ( -. t = (/) -> dom ( t X. t ) = t )
27	26	eqeq1d 2624	. . . . . . . . . 10 |- ( -. t = (/) -> ( dom ( t X. t ) = dom dom G <-> t = dom dom G ) )
28	27	biimpcd 239	. . . . . . . . 9 |- ( dom ( t X. t ) = dom dom G -> ( -. t = (/) -> t = dom dom G ) )
29	28	eqcoms 2630	. . . . . . . 8 |- ( dom dom G = dom ( t X. t ) -> ( -. t = (/) -> t = dom dom G ) )
30	22, 23, 29	3syl 18	. . . . . . 7 |- ( G : ( t X. t ) --> t -> ( -. t = (/) -> t = dom dom G ) )
31	30	com12 32	. . . . . 6 |- ( -. t = (/) -> ( G : ( t X. t ) --> t -> t = dom dom G ) )
32	21, 31	pm2.61i 176	. . . . 5 |- ( G : ( t X. t ) --> t -> t = dom dom G )
33	32	pm4.71ri 665	. . . 4 |- ( G : ( t X. t ) --> t <-> ( t = dom dom G /\ G : ( t X. t ) --> t ) )
34	33	exbii 1774	. . 3 |- ( E. t G : ( t X. t ) --> t <-> E. t ( t = dom dom G /\ G : ( t X. t ) --> t ) )
35	4, 34	syl6bb 276	. 2 |- ( G e. A -> ( G e. Magma <-> E. t ( t = dom dom G /\ G : ( t X. t ) --> t ) ) )
36		dmexg 7097	. . 3 |- ( G e. A -> dom G e. _V )
37		dmexg 7097	. . 3 |- ( dom G e. _V -> dom dom G e. _V )
38		xpeq12 5134	. . . . . . 7 |- ( ( t = dom dom G /\ t = dom dom G ) -> ( t X. t ) = ( dom dom G X. dom dom G ) )
39	38	anidms 677	. . . . . 6 |- ( t = dom dom G -> ( t X. t ) = ( dom dom G X. dom dom G ) )
40		feq23 6029	. . . . . 6 |- ( ( ( t X. t ) = ( dom dom G X. dom dom G ) /\ t = dom dom G ) -> ( G : ( t X. t ) --> t <-> G : ( dom dom G X. dom dom G ) --> dom dom G ) )
41	39, 40	mpancom 703	. . . . 5 |- ( t = dom dom G -> ( G : ( t X. t ) --> t <-> G : ( dom dom G X. dom dom G ) --> dom dom G ) )
42		ismgmOLD.1	. . . . . . . 8 |- X = dom dom G
43	42	eqcomi 2631	. . . . . . 7 |- dom dom G = X
44	43, 43	xpeq12i 5137	. . . . . 6 |- ( dom dom G X. dom dom G ) = ( X X. X )
45	44, 43	feq23i 6039	. . . . 5 |- ( G : ( dom dom G X. dom dom G ) --> dom dom G <-> G : ( X X. X ) --> X )
46	41, 45	syl6bb 276	. . . 4 |- ( t = dom dom G -> ( G : ( t X. t ) --> t <-> G : ( X X. X ) --> X ) )
47	46	ceqsexgv 3335	. . 3 |- ( dom dom G e. _V -> ( E. t ( t = dom dom G /\ G : ( t X. t ) --> t ) <-> G : ( X X. X ) --> X ) )
48	36, 37, 47	3syl 18	. 2 |- ( G e. A -> ( E. t ( t = dom dom G /\ G : ( t X. t ) --> t ) <-> G : ( X X. X ) --> X ) )
49	35, 48	bitrd 268	1 |- ( G e. A -> ( G e. Magma <-> G : ( X X. X ) --> X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   X. cxp 5112   dom cdm 5114   -->wf 5884   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-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-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-mgmOLD 33648

This theorem is referenced by:  clmgmOLD  33650  opidonOLD  33651  issmgrpOLD  33662