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Theorem mhmrcl2 14419
Description: Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
Assertion
Ref Expression
mhmrcl2  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )

Proof of Theorem mhmrcl2
Dummy variables  f 
s  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mhm 14415 . 2  |- MndHom  =  ( s  e.  Mnd , 
t  e.  Mnd  |->  { f  e.  ( (
Base `  t )  ^m  ( Base `  s
) )  |  ( A. x  e.  (
Base `  s ) A. y  e.  ( Base `  s ) ( f `  ( x ( +g  `  s
) y ) )  =  ( ( f `
 x ) ( +g  `  t ) ( f `  y
) )  /\  (
f `  ( 0g `  s ) )  =  ( 0g `  t
) ) } )
21elmpt2cl2 6063 1  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   MndHom cmhm 14413
This theorem is referenced by:  resmhm  14436  mhmco  14439  mhmima  14440  pwsco2mhm  14447  gsumwmhm  14467  mhmmulg  14599  mhmhmeotmd  23300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mhm 14415
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