Theorem grpinvfn 18952

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 7386	. 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V
2		grpinvfn.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2728	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2728	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
5		grpinvfn.n	. . 3 ⊢ 𝑁 = (invg‘𝐺)
6	2, 3, 4, 5	grpinvfval 18949	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
7	1, 6	fnmpti 6703	1 ⊢ 𝑁 Fn 𝐵

Syntax hints:   = wceq 1533   Fn wfn 6548  ‘cfv 6553  ℩crio 7381  (class class class)co 7426  Basecbs 17189  +_gcplusg 17242  0_gc0g 17430  inv_gcminusg 18905

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-riota 7382  df-ov 7429  df-minusg 18908

This theorem is referenced by:  grpinvfvi  18953  isgrpinv  18964  invrfval  20342  mplsubglem  21958  mhpinvcl  22094