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Theorem cinvlem3 25830
Description: The set of the inverses of the morphism  F. (Contributed by FL, 22-Dec-2008.)
Hypotheses
Ref Expression
cinvlem3.1  |-  M  =  dom  ( dom_ `  T
)
cinvlem3.2  |-  D  =  ( dom_ `  T
)
cinvlem3.3  |-  C  =  ( cod_ `  T
)
cinvlem3.4  |-  R  =  ( o_ `  T
)
cinvlem3.5  |-  J  =  ( id_ `  T
)
cinvlem3.6  |-  T  e. 
Cat OLD
cinvlem3.7  |-  F  e.  M
Assertion
Ref Expression
cinvlem3  |-  ( G  e.  ( ( cinv
OLD `  T ) `  F )  <->  ( G  e.  M  /\  ( F R G )  =  ( J `  ( C `  F )
)  /\  ( G R F )  =  ( J `  ( D `
 F ) ) ) )

Proof of Theorem cinvlem3
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . . 5  |-  ( g  =  G  ->  ( F R g )  =  ( F R G ) )
21eqeq1d 2291 . . . 4  |-  ( g  =  G  ->  (
( F R g )  =  ( J `
 ( C `  F ) )  <->  ( F R G )  =  ( J `  ( C `
 F ) ) ) )
3 oveq1 5865 . . . . 5  |-  ( g  =  G  ->  (
g R F )  =  ( G R F ) )
43eqeq1d 2291 . . . 4  |-  ( g  =  G  ->  (
( g R F )  =  ( J `
 ( D `  F ) )  <->  ( G R F )  =  ( J `  ( D `
 F ) ) ) )
52, 4anbi12d 691 . . 3  |-  ( g  =  G  ->  (
( ( F R g )  =  ( J `  ( C `
 F ) )  /\  ( g R F )  =  ( J `  ( D `
 F ) ) )  <->  ( ( F R G )  =  ( J `  ( C `  F )
)  /\  ( G R F )  =  ( J `  ( D `
 F ) ) ) ) )
65elrab 2923 . 2  |-  ( G  e.  { 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 ) ) ) ) )
7 cinvlem3.7 . . . 4  |-  F  e.  M
8 cinvlem3.1 . . . . 5  |-  M  =  dom  ( dom_ `  T
)
9 cinvlem3.2 . . . . 5  |-  D  =  ( dom_ `  T
)
10 cinvlem3.3 . . . . 5  |-  C  =  ( cod_ `  T
)
11 cinvlem3.4 . . . . 5  |-  R  =  ( o_ `  T
)
12 cinvlem3.5 . . . . 5  |-  J  =  ( id_ `  T
)
13 cinvlem3.6 . . . . 5  |-  T  e. 
Cat OLD
148, 9, 10, 11, 12, 13cinvlem2 25829 . . . 4  |-  ( F  e.  M  ->  (
( cinv OLD `  T
) `  F )  =  { g  e.  M  |  ( ( F R g )  =  ( J `  ( C `  F )
)  /\  ( g R F )  =  ( J `  ( D `
 F ) ) ) } )
157, 14ax-mp 8 . . 3  |-  ( (
cinv OLD `  T ) `
 F )  =  { g  e.  M  |  ( ( F R g )  =  ( J `  ( C `  F )
)  /\  ( g R F )  =  ( J `  ( D `
 F ) ) ) }
1615eleq2i 2347 . 2  |-  ( G  e.  ( ( cinv
OLD `  T ) `  F )  <->  G  e.  { g  e.  M  | 
( ( F R g )  =  ( J `  ( C `
 F ) )  /\  ( g R F )  =  ( J `  ( D `
 F ) ) ) } )
17 3anass 938 . 2  |-  ( ( 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 ) ) ) ) )
186, 16, 173bitr4i 268 1  |-  ( G  e.  ( ( 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:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   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 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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