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Theorem mhmrcl1 15780
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 15777 . 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
) ) } )
21elmpt2cl1 6500 1  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   Basecbs 14486   +g cplusg 14551   0gc0g 14691   Mndcmnd 15722   MndHom cmhm 15775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-iota 5549  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-mhm 15777
This theorem is referenced by:  mhmf1o  15786  resmhm2  15801  resmhm2b  15802  mhmco  15803  mhmeql  15805  pwsco2mhm  15812  gsumwmhm  15836  mhmmulg  15974  mhmvlin  18666  mhmhmeotmd  27545
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