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Theorem cinvlem2 25829
Description: The set of the inverses of the morphism  F. (Contributed by FL, 22-Dec-2008.)
Hypotheses
Ref Expression
cinvlem2.1  |-  M  =  dom  ( dom_ `  T
)
cinvlem2.2  |-  D  =  ( dom_ `  T
)
cinvlem2.3  |-  C  =  ( cod_ `  T
)
cinvlem2.4  |-  R  =  ( o_ `  T
)
cinvlem2.5  |-  J  =  ( id_ `  T
)
cinvlem2.6  |-  T  e. 
Cat OLD
Assertion
Ref Expression
cinvlem2  |-  ( F  e.  M  ->  (
( cinv OLD `  T
) `  F )  =  { g  e.  M  |  ( ( F R g )  =  ( J `  ( C `  F )
)  /\  ( g R F )  =  ( J `  ( D `
 F ) ) ) } )
Distinct variable groups:    g, F    g, M    T, g
Allowed substitution hints:    C( g)    D( g)    R( g)    J( g)

Proof of Theorem cinvlem2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 oveq1 5865 . . . . 5  |-  ( f  =  F  ->  (
f R g )  =  ( F R g ) )
2 fveq2 5525 . . . . . 6  |-  ( f  =  F  ->  ( C `  f )  =  ( C `  F ) )
32fveq2d 5529 . . . . 5  |-  ( f  =  F  ->  ( J `  ( C `  f ) )  =  ( J `  ( C `  F )
) )
41, 3eqeq12d 2297 . . . 4  |-  ( f  =  F  ->  (
( f R g )  =  ( J `
 ( C `  f ) )  <->  ( F R g )  =  ( J `  ( C `  F )
) ) )
5 oveq2 5866 . . . . 5  |-  ( f  =  F  ->  (
g R f )  =  ( g R F ) )
6 fveq2 5525 . . . . . 6  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
76fveq2d 5529 . . . . 5  |-  ( f  =  F  ->  ( J `  ( D `  f ) )  =  ( J `  ( D `  F )
) )
85, 7eqeq12d 2297 . . . 4  |-  ( f  =  F  ->  (
( g R f )  =  ( J `
 ( D `  f ) )  <->  ( g R F )  =  ( J `  ( D `
 F ) ) ) )
94, 8anbi12d 691 . . 3  |-  ( f  =  F  ->  (
( ( f R g )  =  ( J `  ( C `
 f ) )  /\  ( g R f )  =  ( J `  ( D `
 f ) ) )  <->  ( ( F R g )  =  ( J `  ( C `  F )
)  /\  ( g R F )  =  ( J `  ( D `
 F ) ) ) ) )
109rabbidv 2780 . 2  |-  ( f  =  F  ->  { g  e.  M  |  ( ( f R g )  =  ( J `
 ( C `  f ) )  /\  ( g R f )  =  ( J `
 ( D `  f ) ) ) }  =  { g  e.  M  |  ( ( F R g )  =  ( J `
 ( C `  F ) )  /\  ( g R F )  =  ( J `
 ( D `  F ) ) ) } )
11 cinvlem2.6 . . 3  |-  T  e. 
Cat OLD
12 cinvlem2.1 . . . 4  |-  M  =  dom  ( dom_ `  T
)
13 cinvlem2.2 . . . 4  |-  D  =  ( dom_ `  T
)
14 cinvlem2.3 . . . 4  |-  C  =  ( cod_ `  T
)
15 cinvlem2.4 . . . 4  |-  R  =  ( o_ `  T
)
16 cinvlem2.5 . . . 4  |-  J  =  ( id_ `  T
)
1712, 13, 14, 15, 16cinvlem1 25828 . . 3  |-  ( 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 )
) ) } ) )
1811, 17ax-mp 8 . 2  |-  ( cinv
OLD `  T )  =  ( f  e.  M  |->  { g  e.  M  |  ( ( f R g )  =  ( J `  ( C `  f ) )  /\  ( g R f )  =  ( J `  ( D `  f )
) ) } )
19 fvex 5539 . . . . 5  |-  ( dom_ `  T )  e.  _V
2019dmex 4941 . . . 4  |-  dom  ( dom_ `  T )  e. 
_V
2112, 20eqeltri 2353 . . 3  |-  M  e. 
_V
2221rabex 4165 . 2  |-  { g  e.  M  |  ( ( F R g )  =  ( J `
 ( C `  F ) )  /\  ( g R F )  =  ( J `
 ( D `  F ) ) ) }  e.  _V
2310, 18, 22fvmpt 5602 1  |-  ( F  e.  M  ->  (
( cinv OLD `  T
) `  F )  =  { 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:  cinvlem3  25830
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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