Theorem grprndm 7988

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

Assertion

Ref	Expression
grprndm	⊢ (G ∈ Grp → ran G = dom dom G)

Proof of Theorem grprndm

Step	Hyp	Ref	Expression
1		eqid 1468	. . 3 ⊢ ran G = ran G
2	1	grpfo 7977	. 2 ⊢ (G ∈ Grp → G:(ran G × ran G)–onto→ran G)
3		fof 3657	. . . . 5 ⊢ (G:(ran G × ran G)–onto→ran G → G:(ran G × ran G)–→ran G)
4		fdm 3617	. . . . 5 ⊢ (G:(ran G × ran G)–→ran G → dom G = (ran G × ran G))
5	3, 4	syl 10	. . . 4 ⊢ (G:(ran G × ran G)–onto→ran G → dom G = (ran G × ran G))
6	5	dmeqd 3302	. . 3 ⊢ (G:(ran G × ran G)–onto→ran G → dom dom G = dom (ran G × ran G))
7		dmxpid 3322	. . 3 ⊢ dom (ran G × ran G) = ran G
8	6, 7	syl6req 1516	. 2 ⊢ (G:(ran G × ran G)–onto→ran G → ran G = dom dom G)
9	2, 8	syl 10	1 ⊢ (G ∈ Grp → ran G = dom dom G)

Syntax hints:   → wi 3   = wceq 953   ∈ wcel 955   × cxp 3158  dom cdm 3160  ran crn 3161  –→wf 3168  –onto→wfo 3170  Grpcgr 7967

This theorem is referenced by:  vcoprne 8136  hhshsslem1 9057

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fo 3186  df-fv 3188  df-opr 3950  df-grp 7971

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