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Theorem cinvlem1 25931
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 5541 . . . . 5  |-  ( t  =  T  ->  ( dom_ `  t )  =  ( dom_ `  T
) )
21dmeqd 4897 . . . 4  |-  ( t  =  T  ->  dom  ( dom_ `  t )  =  dom  ( dom_ `  T
) )
3 cinvlem1.1 . . . 4  |-  M  =  dom  ( dom_ `  T
)
42, 3syl6eqr 2346 . . 3  |-  ( t  =  T  ->  dom  ( dom_ `  t )  =  M )
5 fveq2 5541 . . . . . . . 8  |-  ( t  =  T  ->  (
o_ `  t )  =  ( o_ `  T ) )
6 cinvlem1.4 . . . . . . . 8  |-  R  =  ( o_ `  T
)
75, 6syl6eqr 2346 . . . . . . 7  |-  ( t  =  T  ->  (
o_ `  t )  =  R )
87oveqd 5891 . . . . . 6  |-  ( t  =  T  ->  (
f ( o_ `  t ) g )  =  ( f R g ) )
9 fveq2 5541 . . . . . . . 8  |-  ( t  =  T  ->  ( id_ `  t )  =  ( id_ `  T
) )
10 cinvlem1.5 . . . . . . . 8  |-  J  =  ( id_ `  T
)
119, 10syl6eqr 2346 . . . . . . 7  |-  ( t  =  T  ->  ( id_ `  t )  =  J )
12 fveq2 5541 . . . . . . . . 9  |-  ( t  =  T  ->  ( cod_ `  t )  =  ( cod_ `  T
) )
13 cinvlem1.3 . . . . . . . . 9  |-  C  =  ( cod_ `  T
)
1412, 13syl6eqr 2346 . . . . . . . 8  |-  ( t  =  T  ->  ( cod_ `  t )  =  C )
1514fveq1d 5543 . . . . . . 7  |-  ( t  =  T  ->  (
( cod_ `  t ) `  f )  =  ( C `  f ) )
1611, 15fveq12d 5547 . . . . . 6  |-  ( t  =  T  ->  (
( id_ `  t
) `  ( ( cod_ `  t ) `  f ) )  =  ( J `  ( C `  f )
) )
178, 16eqeq12d 2310 . . . . 5  |-  ( t  =  T  ->  (
( f ( o_
`  t ) g )  =  ( ( id_ `  t ) `
 ( ( cod_ `  t ) `  f
) )  <->  ( f R g )  =  ( J `  ( C `  f )
) ) )
187oveqd 5891 . . . . . 6  |-  ( t  =  T  ->  (
g ( o_ `  t ) f )  =  ( g R f ) )
19 cinvlem1.2 . . . . . . . . 9  |-  D  =  ( dom_ `  T
)
201, 19syl6eqr 2346 . . . . . . . 8  |-  ( t  =  T  ->  ( dom_ `  t )  =  D )
2120fveq1d 5543 . . . . . . 7  |-  ( t  =  T  ->  (
( dom_ `  t ) `  f )  =  ( D `  f ) )
2211, 21fveq12d 5547 . . . . . 6  |-  ( t  =  T  ->  (
( id_ `  t
) `  ( ( dom_ `  t ) `  f ) )  =  ( J `  ( D `  f )
) )
2318, 22eqeq12d 2310 . . . . 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 2796 . . 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 4114 . 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 25930 . 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 5555 . . . . 5  |-  ( dom_ `  T )  e.  _V
2928dmex 4957 . . . 4  |-  dom  ( dom_ `  T )  e. 
_V
303, 29eqeltri 2366 . . 3  |-  M  e. 
_V
3130mptex 5762 . 2  |-  ( f  e.  M  |->  { g  e.  M  |  ( ( f R g )  =  ( J `
 ( C `  f ) )  /\  ( g R f )  =  ( J `
 ( D `  f ) ) ) } )  e.  _V
3226, 27, 31fvmpt 5618 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818    Cat
OLD ccatOLD 25855   cinv
OLDccinv 25929
This theorem is referenced by:  cinvlem2  25932
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-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-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-cinv 25930
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