Theorem resgrprn 8091

Description: The underlying set of a group operation which is a restriction of a mapping. (Contributed by Paul Chapman, 25-Mar-2008.)

Hypothesis

Ref	Expression
resgrprn.1	|- H = (G |` (Y X. Y))

Assertion

Ref	Expression
resgrprn	|- ((dom G = (X X. X) /\ H e. Grp /\ Y (_ X) -> Y = ran H)

Proof of Theorem resgrprn

Step	Hyp	Ref	Expression
1		sseq2 2086	. . . . . . . 8 |- (dom G = (X X. X) -> ((Y X. Y) (_ dom G <-> (Y X. Y) (_ (X X. X)))
2	1	biimpar 419	. . . . . . 7 |- ((dom G = (X X. X) /\ (Y X. Y) (_ (X X. X)) -> (Y X. Y) (_ dom G)
3		ssxp 3262	. . . . . . . 8 |- ((Y (_ X /\ Y (_ X) -> (Y X. Y) (_ (X X. X))
4	3	anidms 436	. . . . . . 7 |- (Y (_ X -> (Y X. Y) (_ (X X. X))
5	2, 4	sylan2 453	. . . . . 6 |- ((dom G = (X X. X) /\ Y (_ X) -> (Y X. Y) (_ dom G)
6		ssdmres 3387	. . . . . 6 |- ((Y X. Y) (_ dom G <-> dom ( G |` (Y X. Y)) = (Y X. Y))
7	5, 6	sylib 198	. . . . 5 |- ((dom G = (X X. X) /\ Y (_ X) -> dom ( G |` (Y X. Y)) = (Y X. Y))
8		resgrprn.1	. . . . . 6 |- H = (G |` (Y X. Y))
9	8	dmeqi 3318	. . . . 5 |- dom H = dom ( G |` (Y X. Y))
10	7, 9	syl5eq 1522	. . . 4 |- ((dom G = (X X. X) /\ Y (_ X) -> dom H = (Y X. Y))
11	10	3adant2 800	. . 3 |- ((dom G = (X X. X) /\ H e. Grp /\ Y (_ X) -> dom H = (Y X. Y))
12		eqid 1478	. . . . . 6 |- ran H = ran H
13	12	grpfo 8040	. . . . 5 |- (H e. Grp -> H:(ran H X. ran H)-onto->ran H)
14		fof 3678	. . . . 5 |- (H:(ran H X. ran H)-onto->ran H -> H:(ran H X. ran H)-->ran H)
15		fdm 3637	. . . . 5 |- (H:(ran H X. ran H)-->ran H -> dom H = (ran H X. ran H))
16	13, 14, 15	3syl 20	. . . 4 |- (H e. Grp -> dom H = (ran H X. ran H))
17	16	3ad2ant2 803	. . 3 |- ((dom G = (X X. X) /\ H e. Grp /\ Y (_ X) -> dom H = (ran H X. ran H))
18	11, 17	eqtr3d 1512	. 2 |- ((dom G = (X X. X) /\ H e. Grp /\ Y (_ X) -> (Y X. Y) = (ran H X. ran H))
19		xpid11 3341	. 2 |- ((Y X. Y) = (ran H X. ran H) <-> Y = ran H)
20	18, 19	sylib 198	1 |- ((dom G = (X X. X) /\ H e. Grp /\ Y (_ X) -> Y = ran H)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   (_ wss 2050   X. cxp 3174  dom cdm 3176  ran crn 3177   |` cres 3178  -->wf 3184  -onto->wfo 3186  Grpcgr 8030

This theorem is referenced by:  ghgrpi 8133  efghgrpilem 8714  shftefif1olem 8736

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  ax-un 2872

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-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-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034

Copyright terms: Public domain