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Theorem grporn 30549
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 2734 . . . . 5 ran 𝐺 = ran 𝐺
32grpofo 30527 . . . 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 6565 . . 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 5945 . . 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 1536  wcel 2105   × cxp 5686  dom cdm 5688  ran crn 5689  Fun wfun 6556   Fn wfn 6557  ontowfo 6560  GrpOpcgr 30517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-ov 7433  df-grpo 30521
This theorem is referenced by:  isabloi  30579  isvciOLD  30608  cnidOLD  30610  cnnv  30705  cnnvba  30707  cncph  30847  hilid  31189  hhnv  31193  hhba  31195  hhph  31206  hhssnv  31292
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