Theorem issubg 8068

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 1649	. . . . . . . . . . 11 |- {h e. Grp | h (_ g} = {h | (h e. Grp /\ h (_ g)}
2		df-pw 2398	. . . . . . . . . . . . 13 |- P~g = {h | h (_ g}
3		visset 1809	. . . . . . . . . . . . . 14 |- g e. V
4	3	pwex 2740	. . . . . . . . . . . . 13 |- P~g e. V
5	2, 4	eqeltrr 1542	. . . . . . . . . . . 12 |- {h | h (_ g} e. V
6		pm3.27 323	. . . . . . . . . . . . 13 |- ((h e. Grp /\ h (_ g) -> h (_ g)
7	6	ss2abi 2116	. . . . . . . . . . . 12 |- {h | (h e. Grp /\ h (_ g)} (_ {h | h (_ g}
8	5, 7	ssexi 2715	. . . . . . . . . . 11 |- {h | (h e. Grp /\ h (_ g)} e. V
9	1, 8	eqeltr 1541	. . . . . . . . . 10 |- {h e. Grp | h (_ g} e. V
10		df-subg 8067	. . . . . . . . . 10 |- SubGrp = {<.g, s>. | (g e. Grp /\ s = {h e. Grp | h (_ g})}
11	9, 10	dmopab2 3611	. . . . . . . . 9 |- dom SubGrp = Grp
12	11	eleq2i 1535	. . . . . . . 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 3736	. . . . . 6 |- (-. G e. dom SubGrp -> (SubGrp` G) = (/))
16		n0i 2281	. . . . . . 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 2742	. . . . . . . . . 10 |- (G e. Grp -> {h | (h (_ G /\ h e. Grp)} e. V)
21		df-rab 1649	. . . . . . . . . . 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 1572	. . . . . . . . . . 11 |- {h | (h e. Grp /\ h (_ G)} = {h | (h (_ G /\ h e. Grp)}
24	21, 23	eqtr 1492	. . . . . . . . . 10 |- {h e. Grp | h (_ G} = {h | (h (_ G /\ h e. Grp)}
25	20, 24	syl5eqel 1549	. . . . . . . . 9 |- (G e. Grp -> {h e. Grp | h (_ G} e. V)
26		sseq2 2079	. . . . . . . . . . 11 |- (g = G -> (h (_ g <-> h (_ G))
27	26	rabbisdv 1803	. . . . . . . . . 10 |- (g = G -> {h e. Grp | h (_ g} = {h e. Grp | h (_ G})
28	27, 10	fvopab4g 3770	. . . . . . . . 9 |- ((G e. Grp /\ {h e. Grp | h (_ G} e. V) -> (SubGrp` G) = {h e. Grp | h (_ G})
29	25, 28	mpdan 703	. . . . . . . 8 |- (G e. Grp -> (SubGrp` G) = {h e. Grp | h (_ G})
30	29	eleq2d 1538	. . . . . . 7 |- (G e. Grp -> (H e. (SubGrp` G) <-> H e. {h e. Grp | h (_ G}))
31		sseq1 2078	. . . . . . . 8 |- (h = H -> (h (_ G <-> H (_ G))
32	31	elrab 1901	. . . . . . 7 |- (H e. {h e. Grp | h (_ G} <-> (H e. Grp /\ H (_ G))
33	30, 32	syl6bb 535	. . . . . 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 778	. . 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 828	. 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 774   = wceq 954   e. wcel 956  {cab 1461  {crab 1645  Vcvv 1807   (_ wss 2043  (/)c0 2276  P~cpw 2397  dom cdm 3165  ` cfv 3177  Grpcgr 7983  SubGrpcsubg 8066

This theorem is referenced by:  subgres 8069  subgid 8072  issubgi 8074  subgabl 8075  ghsubgi 8090  efghgrpilem 8653  hhssabl 9071  ghomfo 10325  ghomgsg 10329  cayleylem3 10345

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-subg 8067

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