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Theorem iddvvidd 25838
Description: Functors preserve domains. JFM CAT1 th. 98. (Contributed by FL, 5-May-2008.)
Hypotheses
Ref Expression
iddvvidd.1  |-  M1  =  dom  ( dom_ `  T
)
iddvvidd.2  |-  D1  =  ( dom_ `  T )
iddvvidd.3  |-  I1  =  ( id_ `  T )
iddvvidd.4  |-  I 2  =  ( id_ `  U
)
iddvvidd.5  |-  D 2  =  ( dom_ `  U
)
Assertion
Ref Expression
iddvvidd  |-  ( ( T  e.  Cat OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  ->  A. m  e.  M1  ( F `  ( I1 `  ( D1 `  m
) ) )  =  ( I 2 `  ( D 2 `  ( F `  m )
) ) ) )
Distinct variable groups:    m, F    m,
M1    T, m    U, m
Allowed substitution hints:    I1( m)    D1( m)    I 2( m)    D 2( m)

Proof of Theorem iddvvidd
Dummy variables  a 
b  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . . 6  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
2 iddvvidd.1 . . . . . 6  |-  M1  =  dom  ( dom_ `  T
)
3 iddvvidd.2 . . . . . 6  |-  D1  =  ( dom_ `  T )
4 eqid 2283 . . . . . 6  |-  ( cod_ `  T )  =  (
cod_ `  T )
5 iddvvidd.3 . . . . . 6  |-  I1  =  ( id_ `  T )
6 eqid 2283 . . . . . 6  |-  ( o_
`  T )  =  ( o_ `  T
)
7 eqid 2283 . . . . . 6  |-  dom  ( id_ `  U )  =  dom  ( id_ `  U
)
8 eqid 2283 . . . . . 6  |-  dom  ( dom_ `  U )  =  dom  ( dom_ `  U
)
9 iddvvidd.5 . . . . . 6  |-  D 2  =  ( dom_ `  U
)
10 eqid 2283 . . . . . 6  |-  ( cod_ `  U )  =  (
cod_ `  U )
11 iddvvidd.4 . . . . . 6  |-  I 2  =  ( id_ `  U
)
12 eqid 2283 . . . . . 6  |-  ( o_
`  U )  =  ( o_ `  U
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12isfunb 25835 . . . . 5  |-  ( ( T  e.  Cat OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  <->  ( F : M1 --> dom  ( dom_ `  U )  /\  ( A. a  e.  dom  ( id_ `  T ) E. b  e.  dom  ( id_ `  U ) ( F `  ( I1 `  a ) )  =  ( I 2 `  b )  /\  ( A. m  e.  M1  ( F `  ( I1 `  ( D1 `  m ) ) )  =  ( I 2 `  ( D 2 `  ( F `
 m ) ) )  /\  A. m  e.  M1  ( F `  ( I1 `  ( (
cod_ `  T ) `  m ) ) )  =  ( I 2 `  ( ( cod_ `  U
) `  ( F `  m ) ) ) )  /\  A. m  e.  M1  A. n  e.  M1  ( ( ( cod_ `  T ) `  n
)  =  ( D1 `  m )  ->  ( F `  ( m
( o_ `  T
) n ) )  =  ( ( F `
 m ) ( o_ `  U ) ( F `  n
) ) ) ) ) ) )
1413simplbda 607 . . . 4  |-  ( ( ( T  e.  Cat OLD 
/\  U  e.  Cat OLD  )  /\  F  e.  ( Func OLD `  <. T ,  U >. )
)  ->  ( A. a  e.  dom  ( id_ `  T ) E. b  e.  dom  ( id_ `  U
) ( F `  ( I1 `  a ) )  =  ( I 2 `  b )  /\  ( A. m  e.  M1  ( F `  ( I1 `  ( D1 `  m ) ) )  =  ( I 2 `  ( D 2 `  ( F `  m ) ) )  /\  A. m  e.  M1  ( F `
 ( I1 `  (
( cod_ `  T ) `  m ) ) )  =  ( I 2 `  ( ( cod_ `  U
) `  ( F `  m ) ) ) )  /\  A. m  e.  M1  A. n  e.  M1  ( ( ( cod_ `  T ) `  n
)  =  ( D1 `  m )  ->  ( F `  ( m
( o_ `  T
) n ) )  =  ( ( F `
 m ) ( o_ `  U ) ( F `  n
) ) ) ) )
1514simp2d 968 . . 3  |-  ( ( ( T  e.  Cat OLD 
/\  U  e.  Cat OLD  )  /\  F  e.  ( Func OLD `  <. T ,  U >. )
)  ->  ( A. m  e.  M1  ( F `
 ( I1 `  ( D1 `  m ) ) )  =  ( I 2 `  ( D 2 `  ( F `
 m ) ) )  /\  A. m  e.  M1  ( F `  ( I1 `  ( (
cod_ `  T ) `  m ) ) )  =  ( I 2 `  ( ( cod_ `  U
) `  ( F `  m ) ) ) ) )
1615simpld 445 . 2  |-  ( ( ( T  e.  Cat OLD 
/\  U  e.  Cat OLD  )  /\  F  e.  ( Func OLD `  <. T ,  U >. )
)  ->  A. m  e.  M1  ( F `  ( I1 `  ( D1 `  m ) ) )  =  ( I 2 `  ( D 2 `  ( F `  m ) ) ) )
1716ex 423 1  |-  ( ( T  e.  Cat OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  ->  A. m  e.  M1  ( F `  ( I1 `  ( D1 `  m
) ) )  =  ( I 2 `  ( D 2 `  ( F `  m )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   <.cop 3643   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715    Cat
OLD ccatOLD 25752   Func
OLDcfuncOLD 25831
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-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-rab 2552  df-v 2790  df-sbc 2992  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-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-funcOLD 25833
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