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Theorem grporndm 27665
Description: A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
Assertion
Ref Expression
grporndm (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)

Proof of Theorem grporndm
StepHypRef Expression
1 eqid 2752 . . 3 ran 𝐺 = ran 𝐺
21grpofo 27654 . 2 (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
3 fof 6268 . . . . 5 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
4 fdm 6204 . . . . 5 (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
53, 4syl 17 . . . 4 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
65dmeqd 5473 . . 3 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺))
7 dmxpid 5492 . . 3 dom (ran 𝐺 × ran 𝐺) = ran 𝐺
86, 7syl6req 2803 . 2 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → ran 𝐺 = dom dom 𝐺)
92, 8syl 17 1 (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  wcel 2131   × cxp 5256  dom cdm 5258  ran crn 5259  wf 6037  ontowfo 6039  GrpOpcgr 27644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fo 6047  df-fv 6049  df-ov 6808  df-grpo 27648
This theorem is referenced by:  hhshsslem1  28425  rngorn1  34037  divrngcl  34061  isdrngo2  34062
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