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Theorem grporndm 28272
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 2821 . . 3 ran 𝐺 = ran 𝐺
21grpofo 28261 . 2 (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
3 fof 6563 . . . . 5 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
43fdmd 6496 . . . 4 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
54dmeqd 5747 . . 3 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺))
6 dmxpid 5773 . . 3 dom (ran 𝐺 × ran 𝐺) = ran 𝐺
75, 6syl6req 2873 . 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 1538  wcel 2115   × cxp 5526  dom cdm 5528  ran crn 5529  ontowfo 6326  GrpOpcgr 28251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-ov 7133  df-grpo 28255
This theorem is referenced by:  hhshsslem1  29029  rngorn1  35257  divrngcl  35281  isdrngo2  35282
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