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Theorem isepia 25922
Description: Epimorphisms of a category  T. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isepia.1  |-  M  =  dom  ( dom_ `  T
)
isepia.2  |-  D  =  ( dom_ `  T
)
isepia.3  |-  C  =  ( cod_ `  T
)
isepia.4  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
isepia  |-  ( 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 ) ) } )
Distinct variable groups:    f, M, g, h    T, f, g, h
Allowed substitution hints:    C( f, g, h)    D( f, g, h)    R( f, g, h)

Proof of Theorem isepia
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5541 . . . . 5  |-  ( x  =  T  ->  ( dom_ `  x )  =  ( dom_ `  T
) )
21dmeqd 4897 . . . 4  |-  ( x  =  T  ->  dom  ( dom_ `  x )  =  dom  ( dom_ `  T
) )
3 isepia.1 . . . 4  |-  M  =  dom  ( dom_ `  T
)
42, 3syl6eqr 2346 . . 3  |-  ( x  =  T  ->  dom  ( dom_ `  x )  =  M )
5 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  T  ->  ( cod_ `  x )  =  ( cod_ `  T
) )
65fveq1d 5543 . . . . . . . . 9  |-  ( x  =  T  ->  (
( cod_ `  x ) `  g )  =  ( ( cod_ `  T
) `  g )
)
7 isepia.3 . . . . . . . . . . . 12  |-  C  =  ( cod_ `  T
)
87eqcomi 2300 . . . . . . . . . . 11  |-  ( cod_ `  T )  =  C
98a1i 10 . . . . . . . . . 10  |-  ( x  =  T  ->  ( cod_ `  T )  =  C )
109fveq1d 5543 . . . . . . . . 9  |-  ( x  =  T  ->  (
( cod_ `  T ) `  g )  =  ( C `  g ) )
116, 10eqtrd 2328 . . . . . . . 8  |-  ( x  =  T  ->  (
( cod_ `  x ) `  g )  =  ( C `  g ) )
125, 7syl6eqr 2346 . . . . . . . . 9  |-  ( x  =  T  ->  ( cod_ `  x )  =  C )
1312fveq1d 5543 . . . . . . . 8  |-  ( x  =  T  ->  (
( cod_ `  x ) `  h )  =  ( C `  h ) )
1411, 13eqeq12d 2310 . . . . . . 7  |-  ( x  =  T  ->  (
( ( cod_ `  x
) `  g )  =  ( ( cod_ `  x ) `  h
)  <->  ( C `  g )  =  ( C `  h ) ) )
15 isepia.2 . . . . . . . . . 10  |-  D  =  ( dom_ `  T
)
161, 15syl6eqr 2346 . . . . . . . . 9  |-  ( x  =  T  ->  ( dom_ `  x )  =  D )
1716fveq1d 5543 . . . . . . . 8  |-  ( x  =  T  ->  (
( dom_ `  x ) `  g )  =  ( D `  g ) )
185fveq1d 5543 . . . . . . . . 9  |-  ( x  =  T  ->  (
( cod_ `  x ) `  f )  =  ( ( cod_ `  T
) `  f )
)
199fveq1d 5543 . . . . . . . . 9  |-  ( x  =  T  ->  (
( cod_ `  T ) `  f )  =  ( C `  f ) )
2018, 19eqtrd 2328 . . . . . . . 8  |-  ( x  =  T  ->  (
( cod_ `  x ) `  f )  =  ( C `  f ) )
2117, 20eqeq12d 2310 . . . . . . 7  |-  ( x  =  T  ->  (
( ( dom_ `  x
) `  g )  =  ( ( cod_ `  x ) `  f
)  <->  ( D `  g )  =  ( C `  f ) ) )
2216fveq1d 5543 . . . . . . . 8  |-  ( x  =  T  ->  (
( dom_ `  x ) `  h )  =  ( D `  h ) )
2322, 20eqeq12d 2310 . . . . . . 7  |-  ( x  =  T  ->  (
( ( dom_ `  x
) `  h )  =  ( ( cod_ `  x ) `  f
)  <->  ( D `  h )  =  ( C `  f ) ) )
2414, 21, 233anbi123d 1252 . . . . . 6  |-  ( x  =  T  ->  (
( ( ( cod_ `  x ) `  g
)  =  ( (
cod_ `  x ) `  h )  /\  (
( dom_ `  x ) `  g )  =  ( ( cod_ `  x
) `  f )  /\  ( ( dom_ `  x
) `  h )  =  ( ( cod_ `  x ) `  f
) )  <->  ( ( C `  g )  =  ( C `  h )  /\  ( D `  g )  =  ( C `  f )  /\  ( D `  h )  =  ( C `  f ) ) ) )
25 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  T  ->  (
o_ `  x )  =  ( o_ `  T ) )
2625oveqd 5891 . . . . . . . . 9  |-  ( x  =  T  ->  (
g ( o_ `  x ) f )  =  ( g ( o_ `  T ) f ) )
27 isepia.4 . . . . . . . . . . . 12  |-  R  =  ( o_ `  T
)
2827eqcomi 2300 . . . . . . . . . . 11  |-  ( o_
`  T )  =  R
2928a1i 10 . . . . . . . . . 10  |-  ( x  =  T  ->  (
o_ `  T )  =  R )
3029oveqd 5891 . . . . . . . . 9  |-  ( x  =  T  ->  (
g ( o_ `  T ) f )  =  ( g R f ) )
3126, 30eqtrd 2328 . . . . . . . 8  |-  ( x  =  T  ->  (
g ( o_ `  x ) f )  =  ( g R f ) )
3225oveqd 5891 . . . . . . . . 9  |-  ( x  =  T  ->  (
h ( o_ `  x ) f )  =  ( h ( o_ `  T ) f ) )
3329oveqd 5891 . . . . . . . . 9  |-  ( x  =  T  ->  (
h ( o_ `  T ) f )  =  ( h R f ) )
3432, 33eqtrd 2328 . . . . . . . 8  |-  ( x  =  T  ->  (
h ( o_ `  x ) f )  =  ( h R f ) )
3531, 34eqeq12d 2310 . . . . . . 7  |-  ( x  =  T  ->  (
( g ( o_
`  x ) f )  =  ( h ( o_ `  x
) f )  <->  ( g R f )  =  ( h R f ) ) )
3635imbi1d 308 . . . . . 6  |-  ( x  =  T  ->  (
( ( g ( o_ `  x ) f )  =  ( h ( o_ `  x ) f )  ->  g  =  h )  <->  ( ( g R f )  =  ( h R f )  ->  g  =  h ) ) )
3724, 36imbi12d 311 . . . . 5  |-  ( x  =  T  ->  (
( ( ( (
cod_ `  x ) `  g )  =  ( ( cod_ `  x
) `  h )  /\  ( ( dom_ `  x
) `  g )  =  ( ( cod_ `  x ) `  f
)  /\  ( ( dom_ `  x ) `  h )  =  ( ( cod_ `  x
) `  f )
)  ->  ( (
g ( o_ `  x ) f )  =  ( h ( o_ `  x ) f )  ->  g  =  h ) )  <->  ( (
( C `  g
)  =  ( C `
 h )  /\  ( D `  g )  =  ( C `  f )  /\  ( D `  h )  =  ( C `  f ) )  -> 
( ( g R f )  =  ( h R f )  ->  g  =  h ) ) ) )
384, 37raleqbidv 2761 . . . 4  |-  ( x  =  T  ->  ( A. h  e.  dom  ( dom_ `  x )
( ( ( (
cod_ `  x ) `  g )  =  ( ( cod_ `  x
) `  h )  /\  ( ( dom_ `  x
) `  g )  =  ( ( cod_ `  x ) `  f
)  /\  ( ( dom_ `  x ) `  h )  =  ( ( cod_ `  x
) `  f )
)  ->  ( (
g ( o_ `  x ) f )  =  ( h ( o_ `  x ) f )  ->  g  =  h ) )  <->  A. h  e.  M  ( (
( C `  g
)  =  ( C `
 h )  /\  ( D `  g )  =  ( C `  f )  /\  ( D `  h )  =  ( C `  f ) )  -> 
( ( g R f )  =  ( h R f )  ->  g  =  h ) ) ) )
394, 38raleqbidv 2761 . . 3  |-  ( x  =  T  ->  ( A. g  e.  dom  ( dom_ `  x ) A. h  e.  dom  ( dom_ `  x )
( ( ( (
cod_ `  x ) `  g )  =  ( ( cod_ `  x
) `  h )  /\  ( ( dom_ `  x
) `  g )  =  ( ( cod_ `  x ) `  f
)  /\  ( ( dom_ `  x ) `  h )  =  ( ( cod_ `  x
) `  f )
)  ->  ( (
g ( o_ `  x ) f )  =  ( h ( o_ `  x ) 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 ) ) ) )
404, 39rabeqbidv 2796 . 2  |-  ( x  =  T  ->  { f  e.  dom  ( dom_ `  x )  |  A. g  e.  dom  ( dom_ `  x ) A. h  e.  dom  ( dom_ `  x
) ( ( ( ( cod_ `  x
) `  g )  =  ( ( cod_ `  x ) `  h
)  /\  ( ( dom_ `  x ) `  g )  =  ( ( cod_ `  x
) `  f )  /\  ( ( dom_ `  x
) `  h )  =  ( ( cod_ `  x ) `  f
) )  ->  (
( g ( o_
`  x ) f )  =  ( h ( o_ `  x
) 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 ) ) } )
41 df-epiOLD 25910 . 2  |- Epic  =  ( x  e.  Cat OLD  |->  { f  e.  dom  ( dom_ `  x )  |  A. g  e.  dom  ( dom_ `  x ) A. h  e.  dom  ( dom_ `  x )
( ( ( (
cod_ `  x ) `  g )  =  ( ( cod_ `  x
) `  h )  /\  ( ( dom_ `  x
) `  g )  =  ( ( cod_ `  x ) `  f
)  /\  ( ( dom_ `  x ) `  h )  =  ( ( cod_ `  x
) `  f )
)  ->  ( (
g ( o_ `  x ) f )  =  ( h ( o_ `  x ) f )  ->  g  =  h ) ) } )
42 fvex 5555 . . . . 5  |-  ( dom_ `  T )  e.  _V
4342dmex 4957 . . . 4  |-  dom  ( dom_ `  T )  e. 
_V
443, 43eqeltri 2366 . . 3  |-  M  e. 
_V
4544rabex 4181 . 2  |-  { 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 ) ) }  e.  _V
4640, 41, 45fvmpt 5618 1  |-  ( 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 ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   dom cdm 4705   ` cfv 5271  (class class class)co 5874   dom_cdom_ 25815   cod_ccod_ 25816   o_co_ 25818    Cat
OLD ccatOLD 25855  EpiccepiOLD 25906
This theorem is referenced by:  isepib  25923
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-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-rab 2565  df-v 2803  df-sbc 3005  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-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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-epiOLD 25910
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