Theorem grporndm 27364

Description: A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Assertion

Ref	Expression
grporndm	|- ( G e. GrpOp -> ran G = dom dom G )

Proof of Theorem grporndm

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ran G = ran G
2	1	grpofo 27353	. 2 |- ( G e. GrpOp -> G : ( ran G X. ran G ) -onto-> ran G )
3		fof 6115	. . . . 5 |- ( G : ( ran G X. ran G ) -onto-> ran G -> G : ( ran G X. ran G ) --> ran G )
4		fdm 6051	. . . . 5 |- ( G : ( ran G X. ran G ) --> ran G -> dom G = ( ran G X. ran G ) )
5	3, 4	syl 17	. . . 4 |- ( G : ( ran G X. ran G ) -onto-> ran G -> dom G = ( ran G X. ran G ) )
6	5	dmeqd 5326	. . 3 |- ( G : ( ran G X. ran G ) -onto-> ran G -> dom dom G = dom ( ran G X. ran G ) )
7		dmxpid 5345	. . 3 |- dom ( ran G X. ran G ) = ran G
8	6, 7	syl6req 2673	. 2 |- ( G : ( ran G X. ran G ) -onto-> ran G -> ran G = dom dom G )
9	2, 8	syl 17	1 |- ( G e. GrpOp -> ran G = dom dom G )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   X. cxp 5112   dom cdm 5114   ran crn 5115   -->wf 5884   -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:  hhshsslem1  28124  rngorn1  33732  divrngcl  33756  isdrngo2  33757