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Theorem grporndm 30530
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 2736 . . 3 ran 𝐺 = ran 𝐺
21grpofo 30519 . 2 (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
3 fof 6819 . . . . 5 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
43fdmd 6745 . . . 4 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
54dmeqd 5915 . . 3 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺))
6 dmxpid 5940 . . 3 dom (ran 𝐺 × ran 𝐺) = ran 𝐺
75, 6eqtr2di 2793 . 2 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → ran 𝐺 = dom dom 𝐺)
82, 7syl 17 1 (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107   × cxp 5682  dom cdm 5684  ran crn 5685  ontowfo 6558  GrpOpcgr 30509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-ov 7435  df-grpo 30513
This theorem is referenced by:  hhshsslem1  31287  rngorn1  37941  divrngcl  37965  isdrngo2  37966
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