Theorem grpinvfn 18923

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 7329	. 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V
2		grpinvfn.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2737	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2737	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
5		grpinvfn.n	. . 3 ⊢ 𝑁 = (invg‘𝐺)
6	2, 3, 4, 5	grpinvfval 18920	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
7	1, 6	fnmpti 6643	1 ⊢ 𝑁 Fn 𝐵

Syntax hints:   = wceq 1542   Fn wfn 6495  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  0_gc0g 17371  inv_gcminusg 18876

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-riota 7325  df-ov 7371  df-minusg 18879

This theorem is referenced by:  grpinvfvi  18924  isgrpinv  18935  invrfval  20337  mplsubglem  21966  mhpinvcl  22107