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Theorem fn0g 18677
Description: The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
Assertion
Ref Expression
fn0g 0g Fn V

Proof of Theorem fn0g
Dummy variables 𝑒 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iotaex 6533 . 2 (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))) ∈ V
2 df-0g 17487 . 2 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))
31, 2fnmpti 6710 1 0g Fn V
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cio 6511   Fn wfn 6555  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  0gc0g 17485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fn 6563  df-0g 17487
This theorem is referenced by:  prdsidlem  18783  pws0g  18787  prdsinvlem  19068  pws1  20323  dsmmbas2  21758  frlmbas  21776
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