Theorem grprndm 8054

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 8043	. 2 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
3		fof 3672	. . . . 5 |- (G:(ran G X. ran G)-onto->ran G -> G:(ran G X. ran G)-->ran G)
4		fdm 3631	. . . . 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 3313	. . 3 |- (G:(ran G X. ran G)-onto->ran G -> dom dom G = dom (ran G X. ran G))
7		dmxpid 3333	. . 3 |- dom (ran G X. ran G) = ran G
8	6, 7	syl6req 1524	. 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 956   e. wcel 958   X. cxp 3168  dom cdm 3170  ran crn 3171  -->wf 3178  -onto->wfo 3180  Grpcgr 8033

This theorem is referenced by:  vcoprne 8198  hhshsslem1 9137

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037

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