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Theorem fmp 25840
Description: Functors preserve morphisms composition. JFM CAT1 th. 99. (Contributed by FL, 2-Aug-2009.)
Hypotheses
Ref Expression
fmp.1  |-  M1  =  dom  ( dom_ `  T
)
fmp.2  |-  C1  =  ( cod_ `  T )
fmp.3  |-  D1  =  ( dom_ `  T )
fmp.6  |-  Ro 1  =  ( o_ `  T )
fmp.7  |-  Ro 2  =  ( o_ `  U )
Assertion
Ref Expression
fmp  |-  ( ( T  e.  Cat OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  ->  A. m  e.  M1  A. n  e.  M1  (
( C1 `  n )  =  ( D1 `  m
)  ->  ( F `  ( m Ro 1
n ) )  =  ( ( F `  m ) Ro 2
( F `  n
) ) ) ) )
Distinct variable groups:    m, n, F    m, M1, n    T, m, n    U, m, n
Allowed substitution hints:    D1( m, n)    C1( m, n)    Ro 1( m, n)    Ro 2( m, n)

Proof of Theorem fmp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . 5  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
2 fmp.1 . . . . 5  |-  M1  =  dom  ( dom_ `  T
)
3 fmp.3 . . . . 5  |-  D1  =  ( dom_ `  T )
4 fmp.2 . . . . 5  |-  C1  =  ( cod_ `  T )
5 eqid 2283 . . . . 5  |-  ( id_ `  T )  =  ( id_ `  T )
6 fmp.6 . . . . 5  |-  Ro 1  =  ( o_ `  T )
7 eqid 2283 . . . . 5  |-  dom  ( id_ `  U )  =  dom  ( id_ `  U
)
8 eqid 2283 . . . . 5  |-  dom  ( dom_ `  U )  =  dom  ( dom_ `  U
)
9 eqid 2283 . . . . 5  |-  ( dom_ `  U )  =  (
dom_ `  U )
10 eqid 2283 . . . . 5  |-  ( cod_ `  U )  =  (
cod_ `  U )
11 eqid 2283 . . . . 5  |-  ( id_ `  U )  =  ( id_ `  U )
12 fmp.7 . . . . 5  |-  Ro 2  =  ( o_ `  U )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12isfunb 25835 . . . 4  |-  ( ( T  e.  Cat OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  <->  ( F : M1 --> dom  ( dom_ `  U )  /\  ( A. x  e.  dom  ( id_ `  T ) E. y  e.  dom  ( id_ `  U ) ( F `  (
( id_ `  T
) `  x )
)  =  ( ( id_ `  U ) `
 y )  /\  ( A. m  e.  M1  ( F `  ( ( id_ `  T ) `
 ( D1 `  m
) ) )  =  ( ( id_ `  U
) `  ( ( dom_ `  U ) `  ( F `  m ) ) )  /\  A. m  e.  M1  ( F `
 ( ( id_ `  T ) `  ( C1 `  m ) ) )  =  ( ( id_ `  U ) `
 ( ( cod_ `  U ) `  ( F `  m )
) ) )  /\  A. m  e.  M1  A. n  e.  M1  ( ( C1 `  n )  =  (
D1 `  m )  ->  ( F `  (
m Ro 1 n
) )  =  ( ( F `  m
) Ro 2 ( F `  n )
) ) ) ) ) )
1413simplbda 607 . . 3  |-  ( ( ( T  e.  Cat OLD 
/\  U  e.  Cat OLD  )  /\  F  e.  ( Func OLD `  <. T ,  U >. )
)  ->  ( A. x  e.  dom  ( id_ `  T ) E. y  e.  dom  ( id_ `  U
) ( F `  ( ( id_ `  T
) `  x )
)  =  ( ( id_ `  U ) `
 y )  /\  ( A. m  e.  M1  ( F `  ( ( id_ `  T ) `
 ( D1 `  m
) ) )  =  ( ( id_ `  U
) `  ( ( dom_ `  U ) `  ( F `  m ) ) )  /\  A. m  e.  M1  ( F `
 ( ( id_ `  T ) `  ( C1 `  m ) ) )  =  ( ( id_ `  U ) `
 ( ( cod_ `  U ) `  ( F `  m )
) ) )  /\  A. m  e.  M1  A. n  e.  M1  ( ( C1 `  n )  =  (
D1 `  m )  ->  ( F `  (
m Ro 1 n
) )  =  ( ( F `  m
) Ro 2 ( F `  n )
) ) ) )
1514simp3d 969 . 2  |-  ( ( ( T  e.  Cat OLD 
/\  U  e.  Cat OLD  )  /\  F  e.  ( Func OLD `  <. T ,  U >. )
)  ->  A. m  e.  M1  A. n  e.  M1  ( ( C1 `  n
)  =  ( D1 `  m )  ->  ( F `  ( m Ro 1 n ) )  =  ( ( F `
 m ) Ro
2 ( F `  n ) ) ) )
1615ex 423 1  |-  ( ( T  e.  Cat OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  ->  A. m  e.  M1  A. n  e.  M1  (
( C1 `  n )  =  ( D1 `  m
)  ->  ( F `  ( m Ro 1
n ) )  =  ( ( F `  m ) Ro 2
( F `  n
) ) ) ) )
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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