Theorem grpinvfn 18621

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 7236	. 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V
2		grpinvfn.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2738	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2738	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
5		grpinvfn.n	. . 3 ⊢ 𝑁 = (invg‘𝐺)
6	2, 3, 4, 5	grpinvfval 18618	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
7	1, 6	fnmpti 6576	1 ⊢ 𝑁 Fn 𝐵

Syntax hints:   = wceq 1539   Fn wfn 6428  ‘cfv 6433  ℩crio 7231  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  0_gc0g 17150  inv_gcminusg 18578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-riota 7232  df-ov 7278  df-minusg 18581

This theorem is referenced by:  grpinvfvi  18622  isgrpinv  18632  invrfval  19915  mplsubglem  21205  mhpinvcl  21342