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Theorem ismona 25809
Description: Monomorphisms of a category. (Contributed by FL, 5-Dec-2007.)
Hypotheses
Ref Expression
ismona.1  |-  M  =  dom  ( dom_ `  T
)
ismona.2  |-  D  =  ( dom_ `  T
)
ismona.3  |-  C  =  ( cod_ `  T
)
ismona.4  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
ismona  |-  ( T  e.  Cat OLD  ->  ( MonoOLD  `  T )  =  {
f  e.  M  |  A. g  e.  M  A. h  e.  M  ( ( ( D `
 g )  =  ( D `  h
)  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) )  ->  ( (
f R g )  =  ( f R h )  ->  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 ismona
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 ismona.1 . . . 4  |-  M  =  dom  ( dom_ `  T
)
42, 3syl6eqr 2333 . . 3  |-  ( x  =  T  ->  dom  ( dom_ `  x )  =  M )
54raleqdv 2742 . . . . 5  |-  ( x  =  T  ->  ( A. h  e.  dom  ( dom_ `  x )
( ( ( (
dom_ `  x ) `  g )  =  ( ( dom_ `  x
) `  h )  /\  ( ( cod_ `  x
) `  g )  =  ( ( dom_ `  x ) `  f
)  /\  ( ( cod_ `  x ) `  h )  =  ( ( dom_ `  x
) `  f )
)  ->  ( (
f ( o_ `  x ) g )  =  ( f ( o_ `  x ) h )  ->  g  =  h ) )  <->  A. h  e.  M  ( (
( ( dom_ `  x
) `  g )  =  ( ( dom_ `  x ) `  h
)  /\  ( ( cod_ `  x ) `  g )  =  ( ( dom_ `  x
) `  f )  /\  ( ( cod_ `  x
) `  h )  =  ( ( dom_ `  x ) `  f
) )  ->  (
( f ( o_
`  x ) g )  =  ( f ( o_ `  x
) h )  -> 
g  =  h ) ) ) )
64, 5raleqbidv 2748 . . . 4  |-  ( x  =  T  ->  ( A. g  e.  dom  ( dom_ `  x ) A. h  e.  dom  ( dom_ `  x )
( ( ( (
dom_ `  x ) `  g )  =  ( ( dom_ `  x
) `  h )  /\  ( ( cod_ `  x
) `  g )  =  ( ( dom_ `  x ) `  f
)  /\  ( ( cod_ `  x ) `  h )  =  ( ( dom_ `  x
) `  f )
)  ->  ( (
f ( o_ `  x ) g )  =  ( f ( o_ `  x ) h )  ->  g  =  h ) )  <->  A. g  e.  M  A. h  e.  M  ( (
( ( dom_ `  x
) `  g )  =  ( ( dom_ `  x ) `  h
)  /\  ( ( cod_ `  x ) `  g )  =  ( ( dom_ `  x
) `  f )  /\  ( ( cod_ `  x
) `  h )  =  ( ( dom_ `  x ) `  f
) )  ->  (
( f ( o_
`  x ) g )  =  ( f ( o_ `  x
) h )  -> 
g  =  h ) ) ) )
71fveq1d 5527 . . . . . . . . 9  |-  ( x  =  T  ->  (
( dom_ `  x ) `  g )  =  ( ( dom_ `  T
) `  g )
)
8 ismona.2 . . . . . . . . . . 11  |-  D  =  ( dom_ `  T
)
98eqcomi 2287 . . . . . . . . . 10  |-  ( dom_ `  T )  =  D
109fveq1i 5526 . . . . . . . . 9  |-  ( (
dom_ `  T ) `  g )  =  ( D `  g )
117, 10syl6eq 2331 . . . . . . . 8  |-  ( x  =  T  ->  (
( dom_ `  x ) `  g )  =  ( D `  g ) )
121fveq1d 5527 . . . . . . . . 9  |-  ( x  =  T  ->  (
( dom_ `  x ) `  h )  =  ( ( dom_ `  T
) `  h )
)
139fveq1i 5526 . . . . . . . . 9  |-  ( (
dom_ `  T ) `  h )  =  ( D `  h )
1412, 13syl6eq 2331 . . . . . . . 8  |-  ( x  =  T  ->  (
( dom_ `  x ) `  h )  =  ( D `  h ) )
1511, 14eqeq12d 2297 . . . . . . 7  |-  ( x  =  T  ->  (
( ( dom_ `  x
) `  g )  =  ( ( dom_ `  x ) `  h
)  <->  ( D `  g )  =  ( D `  h ) ) )
16 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  T  ->  ( cod_ `  x )  =  ( cod_ `  T
) )
1716fveq1d 5527 . . . . . . . . 9  |-  ( x  =  T  ->  (
( cod_ `  x ) `  g )  =  ( ( cod_ `  T
) `  g )
)
18 ismona.3 . . . . . . . . . . 11  |-  C  =  ( cod_ `  T
)
1918eqcomi 2287 . . . . . . . . . 10  |-  ( cod_ `  T )  =  C
2019fveq1i 5526 . . . . . . . . 9  |-  ( (
cod_ `  T ) `  g )  =  ( C `  g )
2117, 20syl6eq 2331 . . . . . . . 8  |-  ( x  =  T  ->  (
( cod_ `  x ) `  g )  =  ( C `  g ) )
221fveq1d 5527 . . . . . . . . 9  |-  ( x  =  T  ->  (
( dom_ `  x ) `  f )  =  ( ( dom_ `  T
) `  f )
)
239fveq1i 5526 . . . . . . . . 9  |-  ( (
dom_ `  T ) `  f )  =  ( D `  f )
2422, 23syl6eq 2331 . . . . . . . 8  |-  ( x  =  T  ->  (
( dom_ `  x ) `  f )  =  ( D `  f ) )
2521, 24eqeq12d 2297 . . . . . . 7  |-  ( x  =  T  ->  (
( ( cod_ `  x
) `  g )  =  ( ( dom_ `  x ) `  f
)  <->  ( C `  g )  =  ( D `  f ) ) )
2616fveq1d 5527 . . . . . . . . 9  |-  ( x  =  T  ->  (
( cod_ `  x ) `  h )  =  ( ( cod_ `  T
) `  h )
)
2719fveq1i 5526 . . . . . . . . 9  |-  ( (
cod_ `  T ) `  h )  =  ( C `  h )
2826, 27syl6eq 2331 . . . . . . . 8  |-  ( x  =  T  ->  (
( cod_ `  x ) `  h )  =  ( C `  h ) )
2928, 24eqeq12d 2297 . . . . . . 7  |-  ( x  =  T  ->  (
( ( cod_ `  x
) `  h )  =  ( ( dom_ `  x ) `  f
)  <->  ( C `  h )  =  ( D `  f ) ) )
3015, 25, 293anbi123d 1252 . . . . . 6  |-  ( x  =  T  ->  (
( ( ( dom_ `  x ) `  g
)  =  ( (
dom_ `  x ) `  h )  /\  (
( cod_ `  x ) `  g )  =  ( ( dom_ `  x
) `  f )  /\  ( ( cod_ `  x
) `  h )  =  ( ( dom_ `  x ) `  f
) )  <->  ( ( D `  g )  =  ( D `  h )  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) ) ) )
31 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  T  ->  (
o_ `  x )  =  ( o_ `  T ) )
3231oveqd 5875 . . . . . . . . 9  |-  ( x  =  T  ->  (
f ( o_ `  x ) g )  =  ( f ( o_ `  T ) g ) )
3331oveqd 5875 . . . . . . . . 9  |-  ( x  =  T  ->  (
f ( o_ `  x ) h )  =  ( f ( o_ `  T ) h ) )
3432, 33eqeq12d 2297 . . . . . . . 8  |-  ( x  =  T  ->  (
( f ( o_
`  x ) g )  =  ( f ( o_ `  x
) h )  <->  ( f
( o_ `  T
) g )  =  ( f ( o_
`  T ) h ) ) )
35 ismona.4 . . . . . . . . . 10  |-  R  =  ( o_ `  T
)
36 oveq 5864 . . . . . . . . . . 11  |-  ( ( o_ `  T )  =  R  ->  (
f ( o_ `  T ) g )  =  ( f R g ) )
3736eqcoms 2286 . . . . . . . . . 10  |-  ( R  =  ( o_ `  T )  ->  (
f ( o_ `  T ) g )  =  ( f R g ) )
3835, 37ax-mp 8 . . . . . . . . 9  |-  ( f ( o_ `  T
) g )  =  ( f R g )
39 oveq 5864 . . . . . . . . . . 11  |-  ( ( o_ `  T )  =  R  ->  (
f ( o_ `  T ) h )  =  ( f R h ) )
4039eqcoms 2286 . . . . . . . . . 10  |-  ( R  =  ( o_ `  T )  ->  (
f ( o_ `  T ) h )  =  ( f R h ) )
4135, 40ax-mp 8 . . . . . . . . 9  |-  ( f ( o_ `  T
) h )  =  ( f R h )
4238, 41eqeq12i 2296 . . . . . . . 8  |-  ( ( f ( o_ `  T ) g )  =  ( f ( o_ `  T ) h )  <->  ( f R g )  =  ( f R h ) )
4334, 42syl6bb 252 . . . . . . 7  |-  ( x  =  T  ->  (
( f ( o_
`  x ) g )  =  ( f ( o_ `  x
) h )  <->  ( f R g )  =  ( f R h ) ) )
4443imbi1d 308 . . . . . 6  |-  ( x  =  T  ->  (
( ( f ( o_ `  x ) g )  =  ( f ( o_ `  x ) h )  ->  g  =  h )  <->  ( ( f R g )  =  ( f R h )  ->  g  =  h ) ) )
4530, 44imbi12d 311 . . . . 5  |-  ( x  =  T  ->  (
( ( ( (
dom_ `  x ) `  g )  =  ( ( dom_ `  x
) `  h )  /\  ( ( cod_ `  x
) `  g )  =  ( ( dom_ `  x ) `  f
)  /\  ( ( cod_ `  x ) `  h )  =  ( ( dom_ `  x
) `  f )
)  ->  ( (
f ( o_ `  x ) g )  =  ( f ( o_ `  x ) h )  ->  g  =  h ) )  <->  ( (
( D `  g
)  =  ( D `
 h )  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) )  -> 
( ( f R g )  =  ( f R h )  ->  g  =  h ) ) ) )
46452ralbidv 2585 . . . 4  |-  ( x  =  T  ->  ( A. g  e.  M  A. h  e.  M  ( ( ( (
dom_ `  x ) `  g )  =  ( ( dom_ `  x
) `  h )  /\  ( ( cod_ `  x
) `  g )  =  ( ( dom_ `  x ) `  f
)  /\  ( ( cod_ `  x ) `  h )  =  ( ( dom_ `  x
) `  f )
)  ->  ( (
f ( o_ `  x ) g )  =  ( f ( o_ `  x ) h )  ->  g  =  h ) )  <->  A. g  e.  M  A. h  e.  M  ( (
( D `  g
)  =  ( D `
 h )  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) )  -> 
( ( f R g )  =  ( f R h )  ->  g  =  h ) ) ) )
476, 46bitrd 244 . . 3  |-  ( x  =  T  ->  ( A. g  e.  dom  ( dom_ `  x ) A. h  e.  dom  ( dom_ `  x )
( ( ( (
dom_ `  x ) `  g )  =  ( ( dom_ `  x
) `  h )  /\  ( ( cod_ `  x
) `  g )  =  ( ( dom_ `  x ) `  f
)  /\  ( ( cod_ `  x ) `  h )  =  ( ( dom_ `  x
) `  f )
)  ->  ( (
f ( o_ `  x ) g )  =  ( f ( o_ `  x ) h )  ->  g  =  h ) )  <->  A. g  e.  M  A. h  e.  M  ( (
( D `  g
)  =  ( D `
 h )  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) )  -> 
( ( f R g )  =  ( f R h )  ->  g  =  h ) ) ) )
484, 47rabeqbidv 2783 . 2  |-  ( x  =  T  ->  { f  e.  dom  ( dom_ `  x )  |  A. g  e.  dom  ( dom_ `  x ) A. h  e.  dom  ( dom_ `  x
) ( ( ( ( dom_ `  x
) `  g )  =  ( ( dom_ `  x ) `  h
)  /\  ( ( cod_ `  x ) `  g )  =  ( ( dom_ `  x
) `  f )  /\  ( ( cod_ `  x
) `  h )  =  ( ( dom_ `  x ) `  f
) )  ->  (
( f ( o_
`  x ) g )  =  ( f ( o_ `  x
) h )  -> 
g  =  h ) ) }  =  {
f  e.  M  |  A. g  e.  M  A. h  e.  M  ( ( ( D `
 g )  =  ( D `  h
)  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) )  ->  ( (
f R g )  =  ( f R h )  ->  g  =  h ) ) } )
49 df-monOLD 25806 . 2  |- MonoOLD  =  ( x  e.  Cat OLD  |->  { f  e.  dom  ( dom_ `  x )  |  A. g  e.  dom  ( dom_ `  x ) A. h  e.  dom  ( dom_ `  x
) ( ( ( ( dom_ `  x
) `  g )  =  ( ( dom_ `  x ) `  h
)  /\  ( ( cod_ `  x ) `  g )  =  ( ( dom_ `  x
) `  f )  /\  ( ( cod_ `  x
) `  h )  =  ( ( dom_ `  x ) `  f
) )  ->  (
( f ( o_
`  x ) g )  =  ( f ( o_ `  x
) h )  -> 
g  =  h ) ) } )
50 fvex 5539 . . . . 5  |-  ( dom_ `  T )  e.  _V
5150dmex 4941 . . . 4  |-  dom  ( dom_ `  T )  e. 
_V
523, 51eqeltri 2353 . . 3  |-  M  e. 
_V
5352rabex 4165 . 2  |-  { f  e.  M  |  A. g  e.  M  A. h  e.  M  (
( ( D `  g )  =  ( D `  h )  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) )  ->  ( (
f R g )  =  ( f R h )  ->  g  =  h ) ) }  e.  _V
5448, 49, 53fvmpt 5602 1  |-  ( T  e.  Cat OLD  ->  ( MonoOLD  `  T )  =  {
f  e.  M  |  A. g  e.  M  A. h  e.  M  ( ( ( D `
 g )  =  ( D `  h
)  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) )  ->  ( (
f R g )  =  ( f R h )  ->  g  =  h ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   dom cdm 4689   ` cfv 5255  (class class class)co 5858   dom_cdom_ 25712   cod_ccod_ 25713   o_co_ 25715    Cat
OLD ccatOLD 25752   MonoOLD cmonOLD 25804
This theorem is referenced by:  ismonb  25810
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-monOLD 25806
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