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Theorem isiso 25825
Description: Isomorphisms of a category. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isiso.1  |-  M  =  dom  ( dom_ `  T
)
isiso.2  |-  D  =  ( dom_ `  T
)
isiso.3  |-  C  =  ( cod_ `  T
)
isiso.4  |-  R  =  ( o_ `  T
)
isiso.5  |-  J  =  ( id_ `  T
)
Assertion
Ref Expression
isiso  |-  ( T  e.  Cat OLD  ->  (  Iso OLD  `  T
)  =  { f  e.  M  |  E. g  e.  M  (
( D `  g
)  =  ( C `
 f )  /\  ( C `  g )  =  ( D `  f )  /\  (
( f R g )  =  ( J `
 ( D `  g ) )  /\  ( 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 isiso
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( x  =  T  ->  ( dom_ `  x )  =  ( dom_ `  T
) )
21dmeqd 4881 . . . 4  |-  ( x  =  T  ->  dom  ( dom_ `  x )  =  dom  ( dom_ `  T
) )
3 isiso.1 . . . 4  |-  M  =  dom  ( dom_ `  T
)
42, 3syl6eqr 2333 . . 3  |-  ( x  =  T  ->  dom  ( dom_ `  x )  =  M )
5 isiso.2 . . . . . . . 8  |-  D  =  ( dom_ `  T
)
61, 5syl6eqr 2333 . . . . . . 7  |-  ( x  =  T  ->  ( dom_ `  x )  =  D )
76fveq1d 5527 . . . . . 6  |-  ( x  =  T  ->  (
( dom_ `  x ) `  g )  =  ( D `  g ) )
8 fveq2 5525 . . . . . . . 8  |-  ( x  =  T  ->  ( cod_ `  x )  =  ( cod_ `  T
) )
9 isiso.3 . . . . . . . 8  |-  C  =  ( cod_ `  T
)
108, 9syl6eqr 2333 . . . . . . 7  |-  ( x  =  T  ->  ( cod_ `  x )  =  C )
1110fveq1d 5527 . . . . . 6  |-  ( x  =  T  ->  (
( cod_ `  x ) `  f )  =  ( C `  f ) )
127, 11eqeq12d 2297 . . . . 5  |-  ( x  =  T  ->  (
( ( dom_ `  x
) `  g )  =  ( ( cod_ `  x ) `  f
)  <->  ( D `  g )  =  ( C `  f ) ) )
1310fveq1d 5527 . . . . . 6  |-  ( x  =  T  ->  (
( cod_ `  x ) `  g )  =  ( C `  g ) )
146fveq1d 5527 . . . . . 6  |-  ( x  =  T  ->  (
( dom_ `  x ) `  f )  =  ( D `  f ) )
1513, 14eqeq12d 2297 . . . . 5  |-  ( x  =  T  ->  (
( ( cod_ `  x
) `  g )  =  ( ( dom_ `  x ) `  f
)  <->  ( C `  g )  =  ( D `  f ) ) )
16 fveq2 5525 . . . . . . . . 9  |-  ( x  =  T  ->  (
o_ `  x )  =  ( o_ `  T ) )
17 isiso.4 . . . . . . . . 9  |-  R  =  ( o_ `  T
)
1816, 17syl6eqr 2333 . . . . . . . 8  |-  ( x  =  T  ->  (
o_ `  x )  =  R )
1918oveqd 5875 . . . . . . 7  |-  ( x  =  T  ->  (
f ( o_ `  x ) g )  =  ( f R g ) )
20 fveq2 5525 . . . . . . . . 9  |-  ( x  =  T  ->  ( id_ `  x )  =  ( id_ `  T
) )
21 isiso.5 . . . . . . . . 9  |-  J  =  ( id_ `  T
)
2220, 21syl6eqr 2333 . . . . . . . 8  |-  ( x  =  T  ->  ( id_ `  x )  =  J )
2322, 7fveq12d 5531 . . . . . . 7  |-  ( x  =  T  ->  (
( id_ `  x
) `  ( ( dom_ `  x ) `  g ) )  =  ( J `  ( D `  g )
) )
2419, 23eqeq12d 2297 . . . . . 6  |-  ( x  =  T  ->  (
( f ( o_
`  x ) g )  =  ( ( id_ `  x ) `
 ( ( dom_ `  x ) `  g
) )  <->  ( f R g )  =  ( J `  ( D `  g )
) ) )
2518oveqd 5875 . . . . . . 7  |-  ( x  =  T  ->  (
g ( o_ `  x ) f )  =  ( g R f ) )
2622, 14fveq12d 5531 . . . . . . 7  |-  ( x  =  T  ->  (
( id_ `  x
) `  ( ( dom_ `  x ) `  f ) )  =  ( J `  ( D `  f )
) )
2725, 26eqeq12d 2297 . . . . . 6  |-  ( x  =  T  ->  (
( g ( o_
`  x ) f )  =  ( ( id_ `  x ) `
 ( ( dom_ `  x ) `  f
) )  <->  ( g R f )  =  ( J `  ( D `  f )
) ) )
2824, 27anbi12d 691 . . . . 5  |-  ( x  =  T  ->  (
( ( f ( o_ `  x ) g )  =  ( ( id_ `  x
) `  ( ( dom_ `  x ) `  g ) )  /\  ( g ( o_
`  x ) f )  =  ( ( id_ `  x ) `
 ( ( dom_ `  x ) `  f
) ) )  <->  ( (
f R g )  =  ( J `  ( D `  g ) )  /\  ( g R f )  =  ( J `  ( D `  f )
) ) ) )
2912, 15, 283anbi123d 1252 . . . 4  |-  ( x  =  T  ->  (
( ( ( dom_ `  x ) `  g
)  =  ( (
cod_ `  x ) `  f )  /\  (
( cod_ `  x ) `  g )  =  ( ( dom_ `  x
) `  f )  /\  ( ( f ( o_ `  x ) g )  =  ( ( id_ `  x
) `  ( ( dom_ `  x ) `  g ) )  /\  ( g ( o_
`  x ) f )  =  ( ( id_ `  x ) `
 ( ( dom_ `  x ) `  f
) ) ) )  <-> 
( ( D `  g )  =  ( C `  f )  /\  ( C `  g )  =  ( D `  f )  /\  ( ( f R g )  =  ( J `  ( D `  g )
)  /\  ( g R f )  =  ( J `  ( D `  f )
) ) ) ) )
304, 29rexeqbidv 2749 . . 3  |-  ( x  =  T  ->  ( E. g  e.  dom  ( dom_ `  x )
( ( ( dom_ `  x ) `  g
)  =  ( (
cod_ `  x ) `  f )  /\  (
( cod_ `  x ) `  g )  =  ( ( dom_ `  x
) `  f )  /\  ( ( f ( o_ `  x ) g )  =  ( ( id_ `  x
) `  ( ( dom_ `  x ) `  g ) )  /\  ( g ( o_
`  x ) f )  =  ( ( id_ `  x ) `
 ( ( dom_ `  x ) `  f
) ) ) )  <->  E. g  e.  M  ( ( D `  g )  =  ( C `  f )  /\  ( C `  g )  =  ( D `  f )  /\  ( ( f R g )  =  ( J `  ( D `  g )
)  /\  ( g R f )  =  ( J `  ( D `  f )
) ) ) ) )
314, 30rabeqbidv 2783 . 2  |-  ( x  =  T  ->  { f  e.  dom  ( dom_ `  x )  |  E. g  e.  dom  ( dom_ `  x ) ( ( ( dom_ `  x
) `  g )  =  ( ( cod_ `  x ) `  f
)  /\  ( ( cod_ `  x ) `  g )  =  ( ( dom_ `  x
) `  f )  /\  ( ( f ( o_ `  x ) g )  =  ( ( id_ `  x
) `  ( ( dom_ `  x ) `  g ) )  /\  ( g ( o_
`  x ) f )  =  ( ( id_ `  x ) `
 ( ( dom_ `  x ) `  f
) ) ) ) }  =  { f  e.  M  |  E. g  e.  M  (
( D `  g
)  =  ( C `
 f )  /\  ( C `  g )  =  ( D `  f )  /\  (
( f R g )  =  ( J `
 ( D `  g ) )  /\  ( g R f )  =  ( J `
 ( D `  f ) ) ) ) } )
32 df-isoc 25808 . 2  |-  Iso OLD  =  ( x  e. 
Cat OLD  |->  { f  e.  dom  ( dom_ `  x )  |  E. g  e.  dom  ( dom_ `  x ) ( ( ( dom_ `  x
) `  g )  =  ( ( cod_ `  x ) `  f
)  /\  ( ( cod_ `  x ) `  g )  =  ( ( dom_ `  x
) `  f )  /\  ( ( f ( o_ `  x ) g )  =  ( ( id_ `  x
) `  ( ( dom_ `  x ) `  g ) )  /\  ( g ( o_
`  x ) f )  =  ( ( id_ `  x ) `
 ( ( dom_ `  x ) `  f
) ) ) ) } )
33 fvex 5539 . . . . 5  |-  ( dom_ `  T )  e.  _V
3433dmex 4941 . . . 4  |-  dom  ( dom_ `  T )  e. 
_V
353, 34eqeltri 2353 . . 3  |-  M  e. 
_V
3635rabex 4165 . 2  |-  { f  e.  M  |  E. g  e.  M  (
( D `  g
)  =  ( C `
 f )  /\  ( C `  g )  =  ( D `  f )  /\  (
( f R g )  =  ( J `
 ( D `  g ) )  /\  ( g R f )  =  ( J `
 ( D `  f ) ) ) ) }  e.  _V
3731, 32, 36fvmpt 5602 1  |-  ( T  e.  Cat OLD  ->  (  Iso OLD  `  T
)  =  { f  e.  M  |  E. g  e.  M  (
( D `  g
)  =  ( C `
 f )  /\  ( C `  g )  =  ( D `  f )  /\  (
( f R g )  =  ( J `
 ( D `  g ) )  /\  ( g R f )  =  ( J `
 ( D `  f ) ) ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788   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    Iso
OLD cisoOLD 25805
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-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-rab 2552  df-v 2790  df-sbc 2992  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-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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-isoc 25808
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