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Theorem grporndm 27492
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 2651 . . 3 ran 𝐺 = ran 𝐺
21grpofo 27481 . 2 (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
3 fof 6153 . . . . 5 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
4 fdm 6089 . . . . 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 5358 . . 3 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺))
7 dmxpid 5377 . . 3 dom (ran 𝐺 × ran 𝐺) = ran 𝐺
86, 7syl6req 2702 . 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 1523  wcel 2030   × cxp 5141  dom cdm 5143  ran crn 5144  wf 5922  ontowfo 5924  GrpOpcgr 27471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-ov 6693  df-grpo 27475
This theorem is referenced by:  hhshsslem1  28252  rngorn1  33862  divrngcl  33886  isdrngo2  33887
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