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Theorem cinvlem1 25828
Description: The set of the inverses of all the morphisms . (Contributed by FL, 22-Dec-2008.)
Hypotheses
Ref Expression
cinvlem1.1  |-  M  =  dom  ( dom_ `  T
)
cinvlem1.2  |-  D  =  ( dom_ `  T
)
cinvlem1.3  |-  C  =  ( cod_ `  T
)
cinvlem1.4  |-  R  =  ( o_ `  T
)
cinvlem1.5  |-  J  =  ( id_ `  T
)
Assertion
Ref Expression
cinvlem1  |-  ( T  e.  Cat OLD  ->  (
cinv OLD `  T )  =  ( f  e.  M  |->  { g  e.  M  |  ( ( f R g )  =  ( J `  ( C `  f ) )  /\  ( g R f )  =  ( J `  ( D `  f )
) ) } ) )
Distinct variable groups:    f, g, M    T, f, g
Allowed substitution hints:    C( f, g)    D( f, g)    R( f, g)    J( f, g)

Proof of Theorem cinvlem1
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( t  =  T  ->  ( dom_ `  t )  =  ( dom_ `  T
) )
21dmeqd 4881 . . . 4  |-  ( t  =  T  ->  dom  ( dom_ `  t )  =  dom  ( dom_ `  T
) )
3 cinvlem1.1 . . . 4  |-  M  =  dom  ( dom_ `  T
)
42, 3syl6eqr 2333 . . 3  |-  ( t  =  T  ->  dom  ( dom_ `  t )  =  M )
5 fveq2 5525 . . . . . . . 8  |-  ( t  =  T  ->  (
o_ `  t )  =  ( o_ `  T ) )
6 cinvlem1.4 . . . . . . . 8  |-  R  =  ( o_ `  T
)
75, 6syl6eqr 2333 . . . . . . 7  |-  ( t  =  T  ->  (
o_ `  t )  =  R )
87oveqd 5875 . . . . . 6  |-  ( t  =  T  ->  (
f ( o_ `  t ) g )  =  ( f R g ) )
9 fveq2 5525 . . . . . . . 8  |-  ( t  =  T  ->  ( id_ `  t )  =  ( id_ `  T
) )
10 cinvlem1.5 . . . . . . . 8  |-  J  =  ( id_ `  T
)
119, 10syl6eqr 2333 . . . . . . 7  |-  ( t  =  T  ->  ( id_ `  t )  =  J )
12 fveq2 5525 . . . . . . . . 9  |-  ( t  =  T  ->  ( cod_ `  t )  =  ( cod_ `  T
) )
13 cinvlem1.3 . . . . . . . . 9  |-  C  =  ( cod_ `  T
)
1412, 13syl6eqr 2333 . . . . . . . 8  |-  ( t  =  T  ->  ( cod_ `  t )  =  C )
1514fveq1d 5527 . . . . . . 7  |-  ( t  =  T  ->  (
( cod_ `  t ) `  f )  =  ( C `  f ) )
1611, 15fveq12d 5531 . . . . . 6  |-  ( t  =  T  ->  (
( id_ `  t
) `  ( ( cod_ `  t ) `  f ) )  =  ( J `  ( C `  f )
) )
178, 16eqeq12d 2297 . . . . 5  |-  ( t  =  T  ->  (
( f ( o_
`  t ) g )  =  ( ( id_ `  t ) `
 ( ( cod_ `  t ) `  f
) )  <->  ( f R g )  =  ( J `  ( C `  f )
) ) )
187oveqd 5875 . . . . . 6  |-  ( t  =  T  ->  (
g ( o_ `  t ) f )  =  ( g R f ) )
19 cinvlem1.2 . . . . . . . . 9  |-  D  =  ( dom_ `  T
)
201, 19syl6eqr 2333 . . . . . . . 8  |-  ( t  =  T  ->  ( dom_ `  t )  =  D )
2120fveq1d 5527 . . . . . . 7  |-  ( t  =  T  ->  (
( dom_ `  t ) `  f )  =  ( D `  f ) )
2211, 21fveq12d 5531 . . . . . 6  |-  ( t  =  T  ->  (
( id_ `  t
) `  ( ( dom_ `  t ) `  f ) )  =  ( J `  ( D `  f )
) )
2318, 22eqeq12d 2297 . . . . 5  |-  ( t  =  T  ->  (
( g ( o_
`  t ) f )  =  ( ( id_ `  t ) `
 ( ( dom_ `  t ) `  f
) )  <->  ( g R f )  =  ( J `  ( D `  f )
) ) )
2417, 23anbi12d 691 . . . 4  |-  ( t  =  T  ->  (
( ( f ( o_ `  t ) g )  =  ( ( id_ `  t
) `  ( ( cod_ `  t ) `  f ) )  /\  ( g ( o_
`  t ) f )  =  ( ( id_ `  t ) `
 ( ( dom_ `  t ) `  f
) ) )  <->  ( (
f R g )  =  ( J `  ( C `  f ) )  /\  ( g R f )  =  ( J `  ( D `  f )
) ) ) )
254, 24rabeqbidv 2783 . . 3  |-  ( t  =  T  ->  { g  e.  dom  ( dom_ `  t )  |  ( ( f ( o_
`  t ) g )  =  ( ( id_ `  t ) `
 ( ( cod_ `  t ) `  f
) )  /\  (
g ( o_ `  t ) f )  =  ( ( id_ `  t ) `  (
( dom_ `  t ) `  f ) ) ) }  =  { g  e.  M  |  ( ( f R g )  =  ( J `
 ( C `  f ) )  /\  ( g R f )  =  ( J `
 ( D `  f ) ) ) } )
264, 25mpteq12dv 4098 . 2  |-  ( t  =  T  ->  (
f  e.  dom  ( dom_ `  t )  |->  { g  e.  dom  ( dom_ `  t )  |  ( ( f ( o_ `  t ) g )  =  ( ( id_ `  t
) `  ( ( cod_ `  t ) `  f ) )  /\  ( g ( o_
`  t ) f )  =  ( ( id_ `  t ) `
 ( ( dom_ `  t ) `  f
) ) ) } )  =  ( f  e.  M  |->  { g  e.  M  |  ( ( f R g )  =  ( J `
 ( C `  f ) )  /\  ( g R f )  =  ( J `
 ( D `  f ) ) ) } ) )
27 df-cinv 25827 . 2  |-  cinv OLD  =  ( t  e. 
Cat OLD  |->  ( f  e.  dom  ( dom_ `  t )  |->  { g  e.  dom  ( dom_ `  t )  |  ( ( f ( o_
`  t ) g )  =  ( ( id_ `  t ) `
 ( ( cod_ `  t ) `  f
) )  /\  (
g ( o_ `  t ) f )  =  ( ( id_ `  t ) `  (
( dom_ `  t ) `  f ) ) ) } ) )
28 fvex 5539 . . . . 5  |-  ( dom_ `  T )  e.  _V
2928dmex 4941 . . . 4  |-  dom  ( dom_ `  T )  e. 
_V
303, 29eqeltri 2353 . . 3  |-  M  e. 
_V
3130mptex 5746 . 2  |-  ( f  e.  M  |->  { g  e.  M  |  ( ( f R g )  =  ( J `
 ( C `  f ) )  /\  ( g R f )  =  ( J `
 ( D `  f ) ) ) } )  e.  _V
3226, 27, 31fvmpt 5602 1  |-  ( T  e.  Cat OLD  ->  (
cinv OLD `  T )  =  ( f  e.  M  |->  { g  e.  M  |  ( ( f R g )  =  ( J `  ( C `  f ) )  /\  ( g R f )  =  ( J `  ( D `  f )
) ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715    Cat
OLD ccatOLD 25752   cinv
OLDccinv 25826
This theorem is referenced by:  cinvlem2  25829
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-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-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-ov 5861  df-cinv 25827
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