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Theorem mhmrcl1 12808
Description: Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
Assertion
Ref Expression
mhmrcl1  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )

Proof of Theorem mhmrcl1
Dummy variables  f  s  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mhm 12805 . 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
) ) } )
21elmpocl1 6069 1  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459   ` cfv 5216  (class class class)co 5874    ^m cmap 6647   Basecbs 12456   +g cplusg 12530   0gc0g 12695   Mndcmnd 12771   MndHom cmhm 12803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-mhm 12805
This theorem is referenced by:  mhmf1o  12815  mhmco  12828  mhmeql  12830  mhmmulg  12977
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