Theorem grporn 27375

Description: The range of a group operation. Useful for satisfying group base set hypotheses of the form X = ran G. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
grporn	|- X = ran G

Proof of Theorem grporn

Step	Hyp	Ref	Expression
1		grprn.1	. . . 4 |- G e. GrpOp
2		eqid 2622	. . . . 5 |- ran G = ran G
3	2	grpofo 27353	. . . 4 |- ( G e. GrpOp -> G : ( ran G X. ran G ) -onto-> ran G )
4		fofun 6116	. . . 4 |- ( G : ( ran G X. ran G ) -onto-> ran G -> Fun G )
5	1, 3, 4	mp2b 10	. . 3 |- Fun G
6		grprn.2	. . 3 |- dom G = ( X X. X )
7		df-fn 5891	. . 3 |- ( G Fn ( X X. X ) <-> ( Fun G /\ dom G = ( X X. X ) ) )
8	5, 6, 7	mpbir2an 955	. 2 |- G Fn ( X X. X )
9		fofn 6117	. . 3 |- ( G : ( ran G X. ran G ) -onto-> ran G -> G Fn ( ran G X. ran G ) )
10	1, 3, 9	mp2b 10	. 2 |- G Fn ( ran G X. ran G )
11		fndmu 5992	. . 3 |- ( ( G Fn ( X X. X ) /\ G Fn ( ran G X. ran G ) ) -> ( X X. X ) = ( ran G X. ran G ) )
12		xpid11 5347	. . 3 |- ( ( X X. X ) = ( ran G X. ran G ) <-> X = ran G )
13	11, 12	sylib 208	. 2 |- ( ( G Fn ( X X. X ) /\ G Fn ( ran G X. ran G ) ) -> X = ran G )
14	8, 10, 13	mp2an 708	1 |- X = ran G

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   X. cxp 5112   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -onto->wfo 5886   GrpOpcgr 27343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-grpo 27347

This theorem is referenced by:  isabloi  27405  isvciOLD  27435  cnidOLD  27437  cnnv  27532  cnnvba  27534  cncph  27674  hilid  28018  hhnv  28022  hhba  28024  hhph  28035  hhssnv  28121