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Theorem fn0g 17868
 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 6334 . 2 (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))) ∈ V
2 df-0g 16710 . 2 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))
31, 2fnmpti 6490 1 0g Fn V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 396   = wceq 1530   ∈ wcel 2107  ∀wral 3143  Vcvv 3500  ℩cio 6311   Fn wfn 6349  ‘cfv 6354  (class class class)co 7150  Basecbs 16478  +gcplusg 16560  0gc0g 16708 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fn 6357  df-0g 16710 This theorem is referenced by:  prdsidlem  17938  pws0g  17942  prdsinvlem  18153  pws1  19302  dsmmbas2  20816  frlmbas  20834
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