Theorem issubg 8112

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 1655	. . . . . . . . . . 11 |- {h e. Grp | h (_ g} = {h | (h e. Grp /\ h (_ g)}
2		df-pw 2406	. . . . . . . . . . . . 13 |- P~g = {h | h (_ g}
3		visset 1816	. . . . . . . . . . . . . 14 |- g e. V
4	3	pwex 2751	. . . . . . . . . . . . 13 |- P~g e. V
5	2, 4	eqeltrr 1548	. . . . . . . . . . . 12 |- {h | h (_ g} e. V
6		pm3.27 323	. . . . . . . . . . . . 13 |- ((h e. Grp /\ h (_ g) -> h (_ g)
7	6	ss2abi 2123	. . . . . . . . . . . 12 |- {h | (h e. Grp /\ h (_ g)} (_ {h | h (_ g}
8	5, 7	ssexi 2725	. . . . . . . . . . 11 |- {h | (h e. Grp /\ h (_ g)} e. V
9	1, 8	eqeltr 1547	. . . . . . . . . 10 |- {h e. Grp | h (_ g} e. V
10		df-subg 8111	. . . . . . . . . 10 |- SubGrp = {<.g, s>. | (g e. Grp /\ s = {h e. Grp | h (_ g})}
11	9, 10	dmopab2 3625	. . . . . . . . 9 |- dom SubGrp = Grp
12	11	eleq2i 1541	. . . . . . . 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 3751	. . . . . 6 |- (-. G e. dom SubGrp -> (SubGrp` G) = (/))
16		n0i 2288	. . . . . . 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 2753	. . . . . . . . . 10 |- (G e. Grp -> {h | (h (_ G /\ h e. Grp)} e. V)
21		df-rab 1655	. . . . . . . . . . 11 |- {h e. Grp | h (_ G} = {h | (h e. Grp /\ h (_ G)}
22		ancom 437	. . . . . . . . . . . 12 |- ((h e. Grp /\ h (_ G) <-> (h (_ G /\ h e. Grp))
23	22	abbii 1578	. . . . . . . . . . 11 |- {h | (h e. Grp /\ h (_ G)} = {h | (h (_ G /\ h e. Grp)}
24	21, 23	eqtr 1498	. . . . . . . . . 10 |- {h e. Grp | h (_ G} = {h | (h (_ G /\ h e. Grp)}
25	20, 24	syl5eqel 1555	. . . . . . . . 9 |- (G e. Grp -> {h e. Grp | h (_ G} e. V)
26		sseq2 2086	. . . . . . . . . . 11 |- (g = G -> (h (_ g <-> h (_ G))
27	26	rabbisdv 1810	. . . . . . . . . 10 |- (g = G -> {h e. Grp | h (_ g} = {h e. Grp | h (_ G})
28	27, 10	fvopab4g 3785	. . . . . . . . 9 |- ((G e. Grp /\ {h e. Grp | h (_ G} e. V) -> (SubGrp` G) = {h e. Grp | h (_ G})
29	25, 28	mpdan 706	. . . . . . . 8 |- (G e. Grp -> (SubGrp` G) = {h e. Grp | h (_ G})
30	29	eleq2d 1544	. . . . . . 7 |- (G e. Grp -> (H e. (SubGrp` G) <-> H e. {h e. Grp | h (_ G}))
31		sseq1 2085	. . . . . . . 8 |- (h = H -> (h (_ G <-> H (_ G))
32	31	elrab 1908	. . . . . . 7 |- (H e. {h e. Grp | h (_ G} <-> (H e. Grp /\ H (_ G))
33	30, 32	syl6bb 538	. . . . . 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 781	. . 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 419	. . 3 |- ((G e. Grp /\ (H e. Grp /\ H (_ G)) -> H e. (SubGrp` G))
40	39	3impb 831	. 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 777   = wceq 958   e. wcel 960  {cab 1466  {crab 1651  Vcvv 1814   (_ wss 2050  (/)c0 2283  P~cpw 2405  dom cdm 3176  ` cfv 3188  Grpcgr 8030  SubGrpcsubg 8110

This theorem is referenced by:  subgres 8113  subgid 8116  issubgi 8118  subgabl 8119  ghsubgi 8134  efghgrpilem 8714  hhssabl 9127  ghomfo 10386  ghomgsg 10390  cayleylem3 10406

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-subg 8111

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