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Theorem isepib 25820
Description: The predicate "is an epimorphism". (Contributed by FL, 8-Aug-2008.)
Hypotheses
Ref Expression
isepib.1  |-  M  =  dom  ( dom_ `  T
)
isepib.2  |-  D  =  ( dom_ `  T
)
isepib.3  |-  C  =  ( cod_ `  T
)
isepib.4  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
isepib  |-  ( T  e.  Cat OLD  ->  ( F  e.  (Epic `  T )  <->  ( F  e.  M  /\  A. g  e.  M  A. h  e.  M  ( (
( C `  g
)  =  ( C `
 h )  /\  ( D `  g )  =  ( C `  F )  /\  ( D `  h )  =  ( C `  F ) )  -> 
( ( g R F )  =  ( h R F )  ->  g  =  h ) ) ) ) )
Distinct variable groups:    g, F, h    g, M, h    T, g, h
Allowed substitution hints:    C( g, h)    D( g, h)    R( g, h)

Proof of Theorem isepib
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 isepib.1 . . . 4  |-  M  =  dom  ( dom_ `  T
)
2 isepib.2 . . . 4  |-  D  =  ( dom_ `  T
)
3 isepib.3 . . . 4  |-  C  =  ( cod_ `  T
)
4 isepib.4 . . . 4  |-  R  =  ( o_ `  T
)
51, 2, 3, 4isepia 25819 . . 3  |-  ( T  e.  Cat OLD  ->  (Epic `  T )  =  {
f  e.  M  |  A. g  e.  M  A. h  e.  M  ( ( ( C `
 g )  =  ( C `  h
)  /\  ( D `  g )  =  ( C `  f )  /\  ( D `  h )  =  ( C `  f ) )  ->  ( (
g R f )  =  ( h R f )  ->  g  =  h ) ) } )
65eleq2d 2350 . 2  |-  ( T  e.  Cat OLD  ->  ( F  e.  (Epic `  T )  <->  F  e.  { f  e.  M  |  A. g  e.  M  A. h  e.  M  ( ( ( C `
 g )  =  ( C `  h
)  /\  ( D `  g )  =  ( C `  f )  /\  ( D `  h )  =  ( C `  f ) )  ->  ( (
g R f )  =  ( h R f )  ->  g  =  h ) ) } ) )
7 fveq2 5525 . . . . . . 7  |-  ( f  =  F  ->  ( C `  f )  =  ( C `  F ) )
87eqeq2d 2294 . . . . . 6  |-  ( f  =  F  ->  (
( D `  g
)  =  ( C `
 f )  <->  ( D `  g )  =  ( C `  F ) ) )
97eqeq2d 2294 . . . . . 6  |-  ( f  =  F  ->  (
( D `  h
)  =  ( C `
 f )  <->  ( D `  h )  =  ( C `  F ) ) )
108, 93anbi23d 1255 . . . . 5  |-  ( f  =  F  ->  (
( ( C `  g )  =  ( C `  h )  /\  ( D `  g )  =  ( C `  f )  /\  ( D `  h )  =  ( C `  f ) )  <->  ( ( C `
 g )  =  ( C `  h
)  /\  ( D `  g )  =  ( C `  F )  /\  ( D `  h )  =  ( C `  F ) ) ) )
11 oveq2 5866 . . . . . . 7  |-  ( f  =  F  ->  (
g R f )  =  ( g R F ) )
12 oveq2 5866 . . . . . . 7  |-  ( f  =  F  ->  (
h R f )  =  ( h R F ) )
1311, 12eqeq12d 2297 . . . . . 6  |-  ( f  =  F  ->  (
( g R f )  =  ( h R f )  <->  ( g R F )  =  ( h R F ) ) )
1413imbi1d 308 . . . . 5  |-  ( f  =  F  ->  (
( ( g R f )  =  ( h R f )  ->  g  =  h )  <->  ( ( g R F )  =  ( h R F )  ->  g  =  h ) ) )
1510, 14imbi12d 311 . . . 4  |-  ( f  =  F  ->  (
( ( ( C `
 g )  =  ( C `  h
)  /\  ( D `  g )  =  ( C `  f )  /\  ( D `  h )  =  ( C `  f ) )  ->  ( (
g R f )  =  ( h R f )  ->  g  =  h ) )  <->  ( (
( C `  g
)  =  ( C `
 h )  /\  ( D `  g )  =  ( C `  F )  /\  ( D `  h )  =  ( C `  F ) )  -> 
( ( g R F )  =  ( h R F )  ->  g  =  h ) ) ) )
16152ralbidv 2585 . . 3  |-  ( f  =  F  ->  ( A. g  e.  M  A. h  e.  M  ( ( ( C `
 g )  =  ( C `  h
)  /\  ( D `  g )  =  ( C `  f )  /\  ( D `  h )  =  ( C `  f ) )  ->  ( (
g R f )  =  ( h R f )  ->  g  =  h ) )  <->  A. g  e.  M  A. h  e.  M  ( (
( C `  g
)  =  ( C `
 h )  /\  ( D `  g )  =  ( C `  F )  /\  ( D `  h )  =  ( C `  F ) )  -> 
( ( g R F )  =  ( h R F )  ->  g  =  h ) ) ) )
1716elrab 2923 . 2  |-  ( F  e.  { f  e.  M  |  A. g  e.  M  A. h  e.  M  ( (
( C `  g
)  =  ( C `
 h )  /\  ( D `  g )  =  ( C `  f )  /\  ( D `  h )  =  ( C `  f ) )  -> 
( ( g R f )  =  ( h R f )  ->  g  =  h ) ) }  <->  ( F  e.  M  /\  A. g  e.  M  A. h  e.  M  ( (
( C `  g
)  =  ( C `
 h )  /\  ( D `  g )  =  ( C `  F )  /\  ( D `  h )  =  ( C `  F ) )  -> 
( ( g R F )  =  ( h R F )  ->  g  =  h ) ) ) )
186, 17syl6bb 252 1  |-  ( T  e.  Cat OLD  ->  ( F  e.  (Epic `  T )  <->  ( F  e.  M  /\  A. g  e.  M  A. h  e.  M  ( (
( C `  g
)  =  ( C `
 h )  /\  ( D `  g )  =  ( C `  F )  /\  ( D `  h )  =  ( C `  F ) )  -> 
( ( g R F )  =  ( h R F )  ->  g  =  h ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   dom cdm 4689   ` cfv 5255  (class class class)co 5858   dom_cdom_ 25712   cod_ccod_ 25713   o_co_ 25715    Cat
OLD ccatOLD 25752  EpiccepiOLD 25803
This theorem is referenced by:  isepib1  25821  isepic  25824
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-epiOLD 25807
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