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Theorem grporndm 30272
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 β†’ ran 𝐺 = dom dom 𝐺)

Proof of Theorem grporndm
StepHypRef Expression
1 eqid 2726 . . 3 ran 𝐺 = ran 𝐺
21grpofo 30261 . 2 (𝐺 ∈ GrpOp β†’ 𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺)
3 fof 6799 . . . . 5 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ 𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
43fdmd 6722 . . . 4 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ dom 𝐺 = (ran 𝐺 Γ— ran 𝐺))
54dmeqd 5899 . . 3 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ dom dom 𝐺 = dom (ran 𝐺 Γ— ran 𝐺))
6 dmxpid 5923 . . 3 dom (ran 𝐺 Γ— ran 𝐺) = ran 𝐺
75, 6eqtr2di 2783 . 2 (𝐺:(ran 𝐺 Γ— ran 𝐺)–ontoβ†’ran 𝐺 β†’ ran 𝐺 = dom dom 𝐺)
82, 7syl 17 1 (𝐺 ∈ GrpOp β†’ ran 𝐺 = dom dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   Γ— cxp 5667  dom cdm 5669  ran crn 5670  β€“ontoβ†’wfo 6535  GrpOpcgr 30251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7408  df-grpo 30255
This theorem is referenced by:  hhshsslem1  31029  rngorn1  37314  divrngcl  37338  isdrngo2  37339
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