Theorem grpinvfn 19000

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 = (invg‘𝐺)

Assertion

Ref	Expression
grpinvfn	⊢ 𝑁 Fn 𝐵

Proof of Theorem grpinvfn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotaex 7393	. 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V
2		grpinvfn.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2736	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2736	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
5		grpinvfn.n	. . 3 ⊢ 𝑁 = (invg‘𝐺)
6	2, 3, 4, 5	grpinvfval 18997	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
7	1, 6	fnmpti 6710	1 ⊢ 𝑁 Fn 𝐵

Syntax hints:   = wceq 1539   Fn wfn 6555  ‘cfv 6560  ℩crio 7388  (class class class)co 7432  Basecbs 17248  +_gcplusg 17298  0_gc0g 17485  inv_gcminusg 18953

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-pow 5364  ax-pr 5431  ax-un 7756

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-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  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-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-riota 7389  df-ov 7435  df-minusg 18956

This theorem is referenced by:  grpinvfvi  19001  isgrpinv  19012  invrfval  20390  mplsubglem  22020  mhpinvcl  22157