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Theorem grporn 27503
 Description: The range of a group operation. Useful for satisfying group base set hypotheses of the form 𝑋 = ran 𝐺. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grprn.1 𝐺 ∈ GrpOp
grprn.2 dom 𝐺 = (𝑋 × 𝑋)
Assertion
Ref Expression
grporn 𝑋 = ran 𝐺

Proof of Theorem grporn
StepHypRef Expression
1 grprn.1 . . . 4 𝐺 ∈ GrpOp
2 eqid 2651 . . . . 5 ran 𝐺 = ran 𝐺
32grpofo 27481 . . . 4 (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
4 fofun 6154 . . . 4 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → Fun 𝐺)
51, 3, 4mp2b 10 . . 3 Fun 𝐺
6 grprn.2 . . 3 dom 𝐺 = (𝑋 × 𝑋)
7 df-fn 5929 . . 3 (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋)))
85, 6, 7mpbir2an 975 . 2 𝐺 Fn (𝑋 × 𝑋)
9 fofn 6155 . . 3 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺𝐺 Fn (ran 𝐺 × ran 𝐺))
101, 3, 9mp2b 10 . 2 𝐺 Fn (ran 𝐺 × ran 𝐺)
11 fndmu 6030 . . 3 ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
12 xpid11 5379 . . 3 ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺)
1311, 12sylib 208 . 2 ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺)
148, 10, 13mp2an 708 1 𝑋 = ran 𝐺
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523   ∈ wcel 2030   × cxp 5141  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  –onto→wfo 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:  isabloi  27533  isvciOLD  27563  cnidOLD  27565  cnnv  27660  cnnvba  27662  cncph  27802  hilid  28146  hhnv  28150  hhba  28152  hhph  28163  hhssnv  28249
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