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Theorem grporn 30540
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 2737 . . . . 5 ran 𝐺 = ran 𝐺
32grpofo 30518 . . . 4 (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
4 fofun 6821 . . . 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 6564 . . 3 (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋)))
85, 6, 7mpbir2an 711 . 2 𝐺 Fn (𝑋 × 𝑋)
9 fofn 6822 . . 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 6675 . . 3 ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
12 xpid11 5943 . . 3 ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺)
1311, 12sylib 218 . 2 ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺)
148, 10, 13mp2an 692 1 𝑋 = ran 𝐺
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108   × cxp 5683  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  ontowfo 6559  GrpOpcgr 30508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7434  df-grpo 30512
This theorem is referenced by:  isabloi  30570  isvciOLD  30599  cnidOLD  30601  cnnv  30696  cnnvba  30698  cncph  30838  hilid  31180  hhnv  31184  hhba  31186  hhph  31197  hhssnv  31283
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