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Theorem grpinvfn 14538
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  =  ( inv g `  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.
StepHypRef Expression
1 riotaex 6324 . 2  |-  ( iota_ y  e.  B ( y ( +g  `  G
) x )  =  ( 0g `  G
) )  e.  _V
2 grpinvfn.b . . 3  |-  B  =  ( Base `  G
)
3 eqid 2296 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2296 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
5 grpinvfn.n . . 3  |-  N  =  ( inv g `  G )
62, 3, 4, 5grpinvfval 14536 . 2  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
71, 6fnmpti 5388 1  |-  N  Fn  B
Colors of variables: wff set class
Syntax hints:    = wceq 1632    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   +g cplusg 13224   0gc0g 13416   inv gcminusg 14379
This theorem is referenced by:  grpinvfvi  14539  isgrpinv  14548  invrfval  15471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-minusg 14506
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