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Theorem mhmrcl2 14744
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 14740 . 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 6292 1  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   ` cfv 5456  (class class class)co 6083    ^m cmap 7020   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Mndcmnd 14686   MndHom cmhm 14738
This theorem is referenced by:  resmhm  14761  mhmco  14764  mhmima  14765  pwsco2mhm  14772  gsumwmhm  14792  mhmmulg  14924  mhmhmeotmd  24315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-dm 4890  df-iota 5420  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-mhm 14740
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