Theorem issubg 8116

Description: The predicate "is a subgroup of G." (Contributed by Paul Chapman, 3-Mar-2008.)

Assertion

Ref	Expression
issubg	|- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H (_ G))

Proof of Theorem issubg

Step	Hyp	Ref	Expression
1		df-rab 1652	. . . . . . . . . . 11 |- {h e. Grp | h (_ g} = {h | (h e. Grp /\ h (_ g)}
2		df-pw 2402	. . . . . . . . . . . . 13 |- P~g = {h | h (_ g}
3		visset 1813	. . . . . . . . . . . . . 14 |- g e. V
4	3	pwex 2745	. . . . . . . . . . . . 13 |- P~g e. V
5	2, 4	eqeltrr 1545	. . . . . . . . . . . 12 |- {h | h (_ g} e. V
6		pm3.27 323	. . . . . . . . . . . . 13 |- ((h e. Grp /\ h (_ g) -> h (_ g)
7	6	ss2abi 2120	. . . . . . . . . . . 12 |- {h | (h e. Grp /\ h (_ g)} (_ {h | h (_ g}
8	5, 7	ssexi 2720	. . . . . . . . . . 11 |- {h | (h e. Grp /\ h (_ g)} e. V
9	1, 8	eqeltr 1544	. . . . . . . . . 10 |- {h e. Grp | h (_ g} e. V
10		df-subg 8115	. . . . . . . . . 10 |- SubGrp = {<.g, s>. | (g e. Grp /\ s = {h e. Grp | h (_ g})}
11	9, 10	dmopab2 3619	. . . . . . . . 9 |- dom SubGrp = Grp
12	11	eleq2i 1538	. . . . . . . 8 |- (G e. dom SubGrp <-> G e. Grp)
13	12	biimp 151	. . . . . . 7 |- (G e. dom SubGrp -> G e. Grp)
14	13	con3i 98	. . . . . 6 |- (-. G e. Grp -> -. G e. dom SubGrp)
15		ndmfv 3745	. . . . . 6 |- (-. G e. dom SubGrp -> (SubGrp` G) = (/))
16		n0i 2285	. . . . . . 7 |- (H e. (SubGrp` G) -> -. (SubGrp` G) = (/))
17	16	con2i 97	. . . . . 6 |- ((SubGrp` G) = (/) -> -. H e. (SubGrp` G))
18	14, 15, 17	3syl 20	. . . . 5 |- (-. G e. Grp -> -. H e. (SubGrp` G))
19	18	a3i 74	. . . 4 |- (H e. (SubGrp` G) -> G e. Grp)
20		abssexg 2747	. . . . . . . . . 10 |- (G e. Grp -> {h | (h (_ G /\ h e. Grp)} e. V)
21		df-rab 1652	. . . . . . . . . . 11 |- {h e. Grp | h (_ G} = {h | (h e. Grp /\ h (_ G)}
22		ancom 435	. . . . . . . . . . . 12 |- ((h e. Grp /\ h (_ G) <-> (h (_ G /\ h e. Grp))
23	22	abbii 1575	. . . . . . . . . . 11 |- {h | (h e. Grp /\ h (_ G)} = {h | (h (_ G /\ h e. Grp)}
24	21, 23	eqtr 1495	. . . . . . . . . 10 |- {h e. Grp | h (_ G} = {h | (h (_ G /\ h e. Grp)}
25	20, 24	syl5eqel 1552	. . . . . . . . 9 |- (G e. Grp -> {h e. Grp | h (_ G} e. V)
26		sseq2 2083	. . . . . . . . . . 11 |- (g = G -> (h (_ g <-> h (_ G))
27	26	rabbisdv 1807	. . . . . . . . . 10 |- (g = G -> {h e. Grp | h (_ g} = {h e. Grp | h (_ G})
28	27, 10	fvopab4g 3779	. . . . . . . . 9 |- ((G e. Grp /\ {h e. Grp | h (_ G} e. V) -> (SubGrp` G) = {h e. Grp | h (_ G})
29	25, 28	mpdan 704	. . . . . . . 8 |- (G e. Grp -> (SubGrp` G) = {h e. Grp | h (_ G})
30	29	eleq2d 1541	. . . . . . 7 |- (G e. Grp -> (H e. (SubGrp` G) <-> H e. {h e. Grp | h (_ G}))
31		sseq1 2082	. . . . . . . 8 |- (h = H -> (h (_ G <-> H (_ G))
32	31	elrab 1905	. . . . . . 7 |- (H e. {h e. Grp | h (_ G} <-> (H e. Grp /\ H (_ G))
33	30, 32	syl6bb 536	. . . . . 6 |- (G e. Grp -> (H e. (SubGrp` G) <-> (H e. Grp /\ H (_ G)))
34	33	biimpd 153	. . . . 5 |- (G e. Grp -> (H e. (SubGrp` G) -> (H e. Grp /\ H (_ G)))
35	19, 34	mpcom 49	. . . 4 |- (H e. (SubGrp` G) -> (H e. Grp /\ H (_ G))
36	19, 35	jca 288	. . 3 |- (H e. (SubGrp` G) -> (G e. Grp /\ (H e. Grp /\ H (_ G)))
37		3anass 779	. . 3 |- ((G e. Grp /\ H e. Grp /\ H (_ G) <-> (G e. Grp /\ (H e. Grp /\ H (_ G)))
38	36, 37	sylibr 200	. 2 |- (H e. (SubGrp` G) -> (G e. Grp /\ H e. Grp /\ H (_ G))
39	33	biimpar 417	. . 3 |- ((G e. Grp /\ (H e. Grp /\ H (_ G)) -> H e. (SubGrp` G))
40	39	3impb 829	. 2 |- ((G e. Grp /\ H e. Grp /\ H (_ G) -> H e. (SubGrp` G))
41	38, 40	impbi 157	1 |- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H (_ G))

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  {cab 1463  {crab 1648  Vcvv 1811   (_ wss 2047  (/)c0 2280  P~cpw 2401  dom cdm 3170  ` cfv 3182  Grpcgr 8033  SubGrpcsubg 8114

This theorem is referenced by:  subgres 8117  subgid 8120  issubgi 8122  subgabl 8123  ghsubgi 8138  efghgrpilem 8719  hhssabl 9132  ghomfo 10391  ghomgsg 10395  cayleylem3 10411

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-subg 8115

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