Theorem fn0g 18646

Description: The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)

Assertion

Ref	Expression
fn0g	⊢ 0g Fn V

Proof of Theorem fn0g

Dummy variables 𝑒 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaex 6509	. 2 ⊢ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))) ∈ V
2		df-0g 17460	. 2 ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
3	1, 2	fnmpti 6686	1 ⊢ 0g Fn V

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464  ℩cio 6487   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  +_gcplusg 17276  0_gc0g 17458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fn 6539  df-0g 17460

This theorem is referenced by:  prdsidlem  18752  pws0g  18756  prdsinvlem  19037  pws1  20290  dsmmbas2  21702  frlmbas  21720