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Theorem cntzmhm2 14815
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 14814 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  x  e.  ( Z `  T
) )  ->  ( F `  x )  e.  ( Y `  ( F " T ) ) )
43ralrimiva 2626 . . 3  |-  ( F  e.  ( G MndHom  H
)  ->  A. x  e.  ( Z `  T
) ( F `  x )  e.  ( Y `  ( F
" T ) ) )
5 ssralv 3237 . . 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 455 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  A. x  e.  S  ( F `  x )  e.  ( Y `  ( F
" T ) ) )
7 eqid 2283 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
8 eqid 2283 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
97, 8mhmf 14420 . . . . 5  |-  ( F  e.  ( G MndHom  H
)  ->  F :
( Base `  G ) --> ( Base `  H )
)
109adantr 451 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  F : ( Base `  G
) --> ( Base `  H
) )
11 ffun 5391 . . . 4  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  Fun  F )
1210, 11syl 15 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  Fun  F )
13 simpr 447 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_  ( Z `  T
) )
147, 1cntzssv 14804 . . . . 5  |-  ( Z `
 T )  C_  ( Base `  G )
1513, 14syl6ss 3191 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_  ( Base `  G
) )
16 fdm 5393 . . . . 5  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  dom  F  =  ( Base `  G
) )
1710, 16syl 15 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  dom  F  =  ( Base `  G
) )
1815, 17sseqtr4d 3215 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_ 
dom  F )
19 funimass4 5573 . . 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 642 . 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 223 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   dom cdm 4689   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   MndHom cmhm 14413  Cntzccntz 14791
This theorem is referenced by:  gsumzmhm  15210  gsumzinv  15217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-mhm 14415  df-cntz 14793
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