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Theorem cntzmhm2 16944
Description: Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
cntzmhm.z  |-  Z  =  (Cntz `  G )
cntzmhm.y  |-  Y  =  (Cntz `  H )
Assertion
Ref Expression
cntzmhm2  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  ( F " S )  C_  ( Y `  ( F
" T ) ) )

Proof of Theorem cntzmhm2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cntzmhm.z . . . . 5  |-  Z  =  (Cntz `  G )
2 cntzmhm.y . . . . 5  |-  Y  =  (Cntz `  H )
31, 2cntzmhm 16943 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  x  e.  ( Z `  T
) )  ->  ( F `  x )  e.  ( Y `  ( F " T ) ) )
43ralrimiva 2846 . . 3  |-  ( F  e.  ( G MndHom  H
)  ->  A. x  e.  ( Z `  T
) ( F `  x )  e.  ( Y `  ( F
" T ) ) )
5 ssralv 3531 . . 3  |-  ( S 
C_  ( Z `  T )  ->  ( A. x  e.  ( Z `  T )
( F `  x
)  e.  ( Y `
 ( F " T ) )  ->  A. x  e.  S  ( F `  x )  e.  ( Y `  ( F " T ) ) ) )
64, 5mpan9 471 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  A. x  e.  S  ( F `  x )  e.  ( Y `  ( F
" T ) ) )
7 eqid 2429 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
8 eqid 2429 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
97, 8mhmf 16538 . . . . 5  |-  ( F  e.  ( G MndHom  H
)  ->  F :
( Base `  G ) --> ( Base `  H )
)
109adantr 466 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  F : ( Base `  G
) --> ( Base `  H
) )
11 ffun 5748 . . . 4  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  Fun  F )
1210, 11syl 17 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  Fun  F )
13 simpr 462 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_  ( Z `  T
) )
147, 1cntzssv 16933 . . . . 5  |-  ( Z `
 T )  C_  ( Base `  G )
1513, 14syl6ss 3482 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_  ( Base `  G
) )
16 fdm 5750 . . . . 5  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  dom  F  =  ( Base `  G
) )
1710, 16syl 17 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  dom  F  =  ( Base `  G
) )
1815, 17sseqtr4d 3507 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_ 
dom  F )
19 funimass4 5932 . . 3  |-  ( ( Fun  F  /\  S  C_ 
dom  F )  -> 
( ( F " S )  C_  ( Y `  ( F " T ) )  <->  A. x  e.  S  ( F `  x )  e.  ( Y `  ( F
" T ) ) ) )
2012, 18, 19syl2anc 665 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  (
( F " S
)  C_  ( Y `  ( F " T
) )  <->  A. x  e.  S  ( F `  x )  e.  ( Y `  ( F
" T ) ) ) )
216, 20mpbird 235 1  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  ( F " S )  C_  ( Y `  ( F
" T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    C_ wss 3442   dom cdm 4854   "cima 4857   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   Basecbs 15084   MndHom cmhm 16531  Cntzccntz 16920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-mhm 16533  df-cntz 16922
This theorem is referenced by:  gsumzmhm  17505  gsumzinv  17513
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