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Theorem cntzmhm2 14831
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 14830 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  x  e.  ( Z `  T
) )  ->  ( F `  x )  e.  ( Y `  ( F " T ) ) )
43ralrimiva 2639 . . 3  |-  ( F  e.  ( G MndHom  H
)  ->  A. x  e.  ( Z `  T
) ( F `  x )  e.  ( Y `  ( F
" T ) ) )
5 ssralv 3250 . . 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 2296 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
8 eqid 2296 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
97, 8mhmf 14436 . . . . 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 5407 . . . 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 14820 . . . . 5  |-  ( Z `
 T )  C_  ( Base `  G )
1513, 14syl6ss 3204 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_  ( Base `  G
) )
16 fdm 5409 . . . . 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 3228 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T
) )  ->  S  C_ 
dom  F )
19 funimass4 5589 . . 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   dom cdm 4705   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   MndHom cmhm 14429  Cntzccntz 14807
This theorem is referenced by:  gsumzmhm  15226  gsumzinv  15233
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-mhm 14431  df-cntz 14809
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