Theorem grprndm 8051

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 1478	. . 3 |- ran G = ran G
2	1	grpfo 8040	. 2 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
3		fof 3678	. . . . 5 |- (G:(ran G X. ran G)-onto->ran G -> G:(ran G X. ran G)-->ran G)
4		fdm 3637	. . . . 5 |- (G:(ran G X. ran G)-->ran G -> dom G = (ran G X. ran G))
5	3, 4	syl 10	. . . 4 |- (G:(ran G X. ran G)-onto->ran G -> dom G = (ran G X. ran G))
6	5	dmeqd 3319	. . 3 |- (G:(ran G X. ran G)-onto->ran G -> dom dom G = dom (ran G X. ran G))
7		dmxpid 3339	. . 3 |- dom (ran G X. ran G) = ran G
8	6, 7	syl6req 1527	. 2 |- (G:(ran G X. ran G)-onto->ran G -> ran G = dom dom G)
9	2, 8	syl 10	1 |- (G e. Grp -> ran G = dom dom G)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960   X. cxp 3174  dom cdm 3176  ran crn 3177  -->wf 3184  -onto->wfo 3186  Grpcgr 8030

This theorem is referenced by:  vcoprne 8194  hhshsslem1 9132

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

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