Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ishoma Unicode version

Theorem ishoma 25787
Description: Definition of  ( hom `  T ). (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
Hypotheses
Ref Expression
ishoma.1  |-  O  =  dom  ( id_ `  T
)
ishoma.2  |-  M  =  dom  ( dom_ `  T
)
ishoma.3  |-  D  =  ( dom_ `  T
)
ishoma.4  |-  C  =  ( cod_ `  T
)
Assertion
Ref Expression
ishoma  |-  ( T  e.  Cat OLD  ->  ( hom `  T )  =  ( a  e.  O ,  b  e.  O  |->  { f  e.  M  |  ( ( D `  f )  =  a  /\  ( C `  f )  =  b ) } ) )
Distinct variable groups:    f, M    a, b, O    f, a, T, b
Allowed substitution hints:    C( f, a, b)    D( f, a, b)    M( a, b)    O( f)

Proof of Theorem ishoma
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( x  =  T  ->  ( id_ `  x )  =  ( id_ `  T
) )
21dmeqd 4881 . . . 4  |-  ( x  =  T  ->  dom  ( id_ `  x )  =  dom  ( id_ `  T ) )
3 ishoma.1 . . . 4  |-  O  =  dom  ( id_ `  T
)
42, 3syl6eqr 2333 . . 3  |-  ( x  =  T  ->  dom  ( id_ `  x )  =  O )
5 fveq2 5525 . . . . . 6  |-  ( x  =  T  ->  ( dom_ `  x )  =  ( dom_ `  T
) )
65dmeqd 4881 . . . . 5  |-  ( x  =  T  ->  dom  ( dom_ `  x )  =  dom  ( dom_ `  T
) )
7 ishoma.2 . . . . 5  |-  M  =  dom  ( dom_ `  T
)
86, 7syl6eqr 2333 . . . 4  |-  ( x  =  T  ->  dom  ( dom_ `  x )  =  M )
9 ishoma.3 . . . . . . . 8  |-  D  =  ( dom_ `  T
)
105, 9syl6eqr 2333 . . . . . . 7  |-  ( x  =  T  ->  ( dom_ `  x )  =  D )
1110fveq1d 5527 . . . . . 6  |-  ( x  =  T  ->  (
( dom_ `  x ) `  f )  =  ( D `  f ) )
1211eqeq1d 2291 . . . . 5  |-  ( x  =  T  ->  (
( ( dom_ `  x
) `  f )  =  a  <->  ( D `  f )  =  a ) )
13 fveq2 5525 . . . . . . . 8  |-  ( x  =  T  ->  ( cod_ `  x )  =  ( cod_ `  T
) )
14 ishoma.4 . . . . . . . 8  |-  C  =  ( cod_ `  T
)
1513, 14syl6eqr 2333 . . . . . . 7  |-  ( x  =  T  ->  ( cod_ `  x )  =  C )
1615fveq1d 5527 . . . . . 6  |-  ( x  =  T  ->  (
( cod_ `  x ) `  f )  =  ( C `  f ) )
1716eqeq1d 2291 . . . . 5  |-  ( x  =  T  ->  (
( ( cod_ `  x
) `  f )  =  b  <->  ( C `  f )  =  b ) )
1812, 17anbi12d 691 . . . 4  |-  ( x  =  T  ->  (
( ( ( dom_ `  x ) `  f
)  =  a  /\  ( ( cod_ `  x
) `  f )  =  b )  <->  ( ( D `  f )  =  a  /\  ( C `  f )  =  b ) ) )
198, 18rabeqbidv 2783 . . 3  |-  ( x  =  T  ->  { f  e.  dom  ( dom_ `  x )  |  ( ( ( dom_ `  x
) `  f )  =  a  /\  (
( cod_ `  x ) `  f )  =  b ) }  =  {
f  e.  M  | 
( ( D `  f )  =  a  /\  ( C `  f )  =  b ) } )
204, 4, 19mpt2eq123dv 5910 . 2  |-  ( x  =  T  ->  (
a  e.  dom  ( id_ `  x ) ,  b  e.  dom  ( id_ `  x )  |->  { f  e.  dom  ( dom_ `  x )  |  ( ( ( dom_ `  x ) `  f
)  =  a  /\  ( ( cod_ `  x
) `  f )  =  b ) } )  =  ( a  e.  O ,  b  e.  O  |->  { f  e.  M  |  ( ( D `  f
)  =  a  /\  ( C `  f )  =  b ) } ) )
21 df-homOLD 25786 . 2  |-  hom  =  ( x  e.  Cat OLD  |->  ( a  e.  dom  ( id_ `  x ) ,  b  e.  dom  ( id_ `  x ) 
|->  { f  e.  dom  ( dom_ `  x )  |  ( ( (
dom_ `  x ) `  f )  =  a  /\  ( ( cod_ `  x ) `  f
)  =  b ) } ) )
22 fvex 5539 . . . . 5  |-  ( id_ `  T )  e.  _V
2322dmex 4941 . . . 4  |-  dom  ( id_ `  T )  e. 
_V
243, 23eqeltri 2353 . . 3  |-  O  e. 
_V
2524, 24mpt2ex 6198 . 2  |-  ( a  e.  O ,  b  e.  O  |->  { f  e.  M  |  ( ( D `  f
)  =  a  /\  ( C `  f )  =  b ) } )  e.  _V
2620, 21, 25fvmpt 5602 1  |-  ( T  e.  Cat OLD  ->  ( hom `  T )  =  ( a  e.  O ,  b  e.  O  |->  { f  e.  M  |  ( ( D `  f )  =  a  /\  ( C `  f )  =  b ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   dom cdm 4689   ` cfv 5255    e. cmpt2 5860   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714    Cat
OLD ccatOLD 25752   homchomOLD 25785
This theorem is referenced by:  ishomb  25788
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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-homOLD 25786
  Copyright terms: Public domain W3C validator