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Theorem grpinvfn 18145
 Description: Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
grpinvfn.b 𝐵 = (Base‘𝐺)
grpinvfn.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvfn 𝑁 Fn 𝐵

Proof of Theorem grpinvfn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 7111 . 2 (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ V
2 grpinvfn.b . . 3 𝐵 = (Base‘𝐺)
3 eqid 2824 . . 3 (+g𝐺) = (+g𝐺)
4 eqid 2824 . . 3 (0g𝐺) = (0g𝐺)
5 grpinvfn.n . . 3 𝑁 = (invg𝐺)
62, 3, 4, 5grpinvfval 18142 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)))
71, 6fnmpti 6480 1 𝑁 Fn 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   Fn wfn 6338  ‘cfv 6343  ℩crio 7106  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  0gc0g 16713  invgcminusg 18104 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-riota 7107  df-ov 7152  df-minusg 18107 This theorem is referenced by:  grpinvfvi  18146  isgrpinv  18156  invrfval  19426
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