Theorem grprndm 9334

Description: A group's range in terms of its domain.

Assertion

Ref	Expression
grprndm	|- (G e. Grp -> ran G = dom dom G)

Proof of Theorem grprndm

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- ran G = ran G
2	1	grpfo 9323	. 2 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
3		fof 4617	. . . . 5 |- (G:(ran G X. ran G)-onto->ran G -> G:(ran G X. ran G)-->ran G)
4		fdm 4567	. . . . 5 |- (G:(ran G X. ran G)-->ran G -> dom G = (ran G X. ran G))
5	3, 4	syl 12	. . . 4 |- (G:(ran G X. ran G)-onto->ran G -> dom G = (ran G X. ran G))
6	5	dmeqd 4159	. . 3 |- (G:(ran G X. ran G)-onto->ran G -> dom dom G = dom (ran G X. ran G))
7		dmxpid 4179	. . 3 |- dom (ran G X. ran G) = ran G
8	6, 7	syl6req 1945	. 2 |- (G:(ran G X. ran G)-onto->ran G -> ran G = dom dom G)
9	2, 8	syl 12	1 |- (G e. Grp -> ran G = dom dom G)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300   X. cxp 3984  dom cdm 3986  ran crn 3987  -->wf 3994  -onto->wfo 3996  Grpcgr 9311

This theorem is referenced by:  vcoprne 9530  rnplrnml2 10403  hhshsslem1 10770  ablcomgrp 14702  ltlga 14729  fprodsub 14742  prsubrtr 14763  tpgprop2 14987  isablda 16035  pi1set 16096  divrngcl 16110  isdivrng2 16111

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-grp 9316

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