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Theorem cinvlem2 25932
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 5881 . . . . 5  |-  ( f  =  F  ->  (
f R g )  =  ( F R g ) )
2 fveq2 5541 . . . . . 6  |-  ( f  =  F  ->  ( C `  f )  =  ( C `  F ) )
32fveq2d 5545 . . . . 5  |-  ( f  =  F  ->  ( J `  ( C `  f ) )  =  ( J `  ( C `  F )
) )
41, 3eqeq12d 2310 . . . 4  |-  ( f  =  F  ->  (
( f R g )  =  ( J `
 ( C `  f ) )  <->  ( F R g )  =  ( J `  ( C `  F )
) ) )
5 oveq2 5882 . . . . 5  |-  ( f  =  F  ->  (
g R f )  =  ( g R F ) )
6 fveq2 5541 . . . . . 6  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
76fveq2d 5545 . . . . 5  |-  ( f  =  F  ->  ( J `  ( D `  f ) )  =  ( J `  ( D `  F )
) )
85, 7eqeq12d 2310 . . . 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 2793 . 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 25931 . . 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 5555 . . . . 5  |-  ( dom_ `  T )  e.  _V
2019dmex 4957 . . . 4  |-  dom  ( dom_ `  T )  e. 
_V
2112, 20eqeltri 2366 . . 3  |-  M  e. 
_V
2221rabex 4181 . 2  |-  { g  e.  M  |  ( ( F R g )  =  ( J `
 ( C `  F ) )  /\  ( g R F )  =  ( J `
 ( D `  F ) ) ) }  e.  _V
2310, 18, 22fvmpt 5618 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 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:  cinvlem3  25933
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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