Theorem grprndm 8037

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 1475	. . 3 |- ran G = ran G
2	1	grpfo 8026	. 2 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
3		fof 3669	. . . . 5 |- (G:(ran G X. ran G)-onto->ran G -> G:(ran G X. ran G)-->ran G)
4		fdm 3628	. . . . 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 3310	. . 3 |- (G:(ran G X. ran G)-onto->ran G -> dom dom G = dom (ran G X. ran G))
7		dmxpid 3330	. . 3 |- dom (ran G X. ran G) = ran G
8	6, 7	syl6req 1523	. 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 955   e. wcel 957   X. cxp 3165  dom cdm 3167  ran crn 3168  -->wf 3175  -onto->wfo 3177  Grpcgr 8016

This theorem is referenced by:  vcoprne 8183  hhshsslem1 9125

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fo 3193  df-fv 3195  df-opr 3962  df-grp 8020

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