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Theorem cntzmhm2 15140
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 15139 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  x  e.  ( Z `  T
) )  ->  ( F `  x )  e.  ( Y `  ( F " T ) ) )
43ralrimiva 2791 . . 3  |-  ( F  e.  ( G MndHom  H
)  ->  A. x  e.  ( Z `  T
) ( F `  x )  e.  ( Y `  ( F
" T ) ) )
5 ssralv 3409 . . 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 457 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  A. x  e.  S  ( F `  x )  e.  ( Y `  ( F
" T ) ) )
7 eqid 2438 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
8 eqid 2438 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
97, 8mhmf 14745 . . . . 5  |-  ( F  e.  ( G MndHom  H
)  ->  F :
( Base `  G ) --> ( Base `  H )
)
109adantr 453 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  F : ( Base `  G
) --> ( Base `  H
) )
11 ffun 5595 . . . 4  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  Fun  F )
1210, 11syl 16 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  Fun  F )
13 simpr 449 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_  ( Z `  T
) )
147, 1cntzssv 15129 . . . . 5  |-  ( Z `
 T )  C_  ( Base `  G )
1513, 14syl6ss 3362 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_  ( Base `  G
) )
16 fdm 5597 . . . . 5  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  dom  F  =  ( Base `  G
) )
1710, 16syl 16 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  dom  F  =  ( Base `  G
) )
1815, 17sseqtr4d 3387 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_ 
dom  F )
19 funimass4 5779 . . 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 644 . 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 225 1  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  ( F " S )  C_  ( Y `  ( F
" T ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   dom cdm 4880   "cima 4883   Fun wfun 5450   -->wf 5452   ` cfv 5456  (class class class)co 6083   Basecbs 13471   MndHom cmhm 14738  Cntzccntz 15116
This theorem is referenced by:  gsumzmhm  15535  gsumzinv  15542
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-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-mhm 14740  df-cntz 15118
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