Theorem grprn 8056

Description: The range of a group operation. Useful for satisfying group base set hypotheses of the form X = ran G.

Hypotheses

Ref	Expression
grprn.1	|- G e. Grp
grprn.2	|- dom G = (X X. X)

Assertion

Ref	Expression
grprn	|- X = ran G

Proof of Theorem grprn

Step	Hyp	Ref	Expression
1		df-fn 3193	. . 3 |- (G Fn (X X. X) <-> (Fun G /\ dom G = (X X. X)))
2		grprn.1	. . . . 5 |- G e. Grp
3		eqid 1475	. . . . . 6 |- ran G = ran G
4	3	grpfo 8043	. . . . 5 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
5	2, 4	ax-mp 7	. . . 4 |- G:(ran G X. ran G)-onto->ran G
6		fofun 3673	. . . 4 |- (G:(ran G X. ran G)-onto->ran G -> Fun G)
7	5, 6	ax-mp 7	. . 3 |- Fun G
8		grprn.2	. . 3 |- dom G = (X X. X)
9	1, 7, 8	mpbir2an 730	. 2 |- G Fn (X X. X)
10		fof 3672	. . . 4 |- (G:(ran G X. ran G)-onto->ran G -> G:(ran G X. ran G)-->ran G)
11	5, 10	ax-mp 7	. . 3 |- G:(ran G X. ran G)-->ran G
12		ffn 3627	. . 3 |- (G:(ran G X. ran G)-->ran G -> G Fn (ran G X. ran G))
13	11, 12	ax-mp 7	. 2 |- G Fn (ran G X. ran G)
14		fndmu 3589	. . 3 |- ((G Fn (X X. X) /\ G Fn (ran G X. ran G)) -> (X X. X) = (ran G X. ran G))
15		xpid11 3335	. . 3 |- ((X X. X) = (ran G X. ran G) <-> X = ran G)
16	14, 15	sylib 198	. 2 |- ((G Fn (X X. X) /\ G Fn (ran G X. ran G)) -> X = ran G)
17	9, 13, 16	mp2an 697	1 |- X = ran G

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958   X. cxp 3168  dom cdm 3170  ran crn 3171  Fun wfun 3176   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  Grpcgr 8033

This theorem is referenced by:  grprnOLD 8057  isabli 8106  cnid 8127  addinv 8128  readdsubg 8129  zaddsubg 8130  mulid 8132  cnring 8162  isvci 8201  cnnv 8307  cnnvba 8309  cnph 8478  efghgrpilem 8719  hilid 9028  hhnv 9032  hhba 9034  hhph 9045  hhssnv 9134  symgidi 10407  cayleylem2 10410

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  ax-un 2866

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-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-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037

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