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Theorem cntzmhm2 16226
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 16225 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  x  e.  ( Z `  T
) )  ->  ( F `  x )  e.  ( Y `  ( F " T ) ) )
43ralrimiva 2881 . . 3  |-  ( F  e.  ( G MndHom  H
)  ->  A. x  e.  ( Z `  T
) ( F `  x )  e.  ( Y `  ( F
" T ) ) )
5 ssralv 3569 . . 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 469 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  A. x  e.  S  ( F `  x )  e.  ( Y `  ( F
" T ) ) )
7 eqid 2467 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
8 eqid 2467 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
97, 8mhmf 15824 . . . . 5  |-  ( F  e.  ( G MndHom  H
)  ->  F :
( Base `  G ) --> ( Base `  H )
)
109adantr 465 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  F : ( Base `  G
) --> ( Base `  H
) )
11 ffun 5738 . . . 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 461 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_  ( Z `  T
) )
147, 1cntzssv 16215 . . . . 5  |-  ( Z `
 T )  C_  ( Base `  G )
1513, 14syl6ss 3521 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_  ( Base `  G
) )
16 fdm 5740 . . . . 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 3546 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_ 
dom  F )
19 funimass4 5924 . . 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 661 . 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 232 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817    C_ wss 3481   dom cdm 5004   "cima 5007   Fun wfun 5587   -->wf 5589   ` cfv 5593  (class class class)co 6294   Basecbs 14502   MndHom cmhm 15817  Cntzccntz 16202
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-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-map 7432  df-mhm 15819  df-cntz 16204
This theorem is referenced by:  gsumzmhm  16807  gsumzmhmOLD  16808  gsumzinv  16819  gsumzinvOLD  16820
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