MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grporn Structured version   Visualization version   GIF version

Theorem grporn 29292
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 2738 . . . . 5 ran 𝐺 = ran 𝐺
32grpofo 29270 . . . 4 (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
4 fofun 6755 . . . 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 6497 . . 3 (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋)))
85, 6, 7mpbir2an 710 . 2 𝐺 Fn (𝑋 × 𝑋)
9 fofn 6756 . . 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 6607 . . 3 ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
12 xpid11 5886 . . 3 ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺)
1311, 12sylib 217 . 2 ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺)
148, 10, 13mp2an 691 1 𝑋 = ran 𝐺
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107   × cxp 5630  dom cdm 5632  ran crn 5633  Fun wfun 6488   Fn wfn 6489  ontowfo 6492  GrpOpcgr 29260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fo 6500  df-fv 6502  df-ov 7355  df-grpo 29264
This theorem is referenced by:  isabloi  29322  isvciOLD  29351  cnidOLD  29353  cnnv  29448  cnnvba  29450  cncph  29590  hilid  29932  hhnv  29936  hhba  29938  hhph  29949  hhssnv  30035
  Copyright terms: Public domain W3C validator