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Theorem mhmrcl1 14434
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 14431 . 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 6078 1  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   MndHom cmhm 14429
This theorem is referenced by:  resmhm2  14453  resmhm2b  14454  mhmco  14455  mhmeql  14457  pwsco2mhm  14463  gsumwmhm  14483  mhmmulg  14615  mhmhmeotmd  23315  mhmvlin  27555
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-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-rab 2565  df-v 2803  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-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-iota 5235  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-mhm 14431
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