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Theorem ismonb 25913
Description: The predicate "is a monomorphism". (Contributed by FL, 2-Jan-2008.)
Hypotheses
Ref Expression
ismonb.1  |-  M  =  dom  ( dom_ `  T
)
ismonb.2  |-  D  =  ( dom_ `  T
)
ismonb.3  |-  C  =  ( cod_ `  T
)
ismonb.4  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
ismonb  |-  ( T  e.  Cat OLD  ->  ( F  e.  ( 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:    g, F, h    g, M, h    T, g, h
Allowed substitution hints:    C( g, h)    D( g, h)    R( g, h)

Proof of Theorem ismonb
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ismonb.1 . . . 4  |-  M  =  dom  ( dom_ `  T
)
2 ismonb.2 . . . 4  |-  D  =  ( dom_ `  T
)
3 ismonb.3 . . . 4  |-  C  =  ( cod_ `  T
)
4 ismonb.4 . . . 4  |-  R  =  ( o_ `  T
)
51, 2, 3, 4ismona 25912 . . 3  |-  ( 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 ) ) } )
65eleq2d 2363 . 2  |-  ( T  e.  Cat OLD  ->  ( F  e.  ( MonoOLD  `  T
)  <->  F  e.  { 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 ) ) } ) )
7 fveq2 5541 . . . . . . 7  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
87eqeq2d 2307 . . . . . 6  |-  ( f  =  F  ->  (
( C `  g
)  =  ( D `
 f )  <->  ( C `  g )  =  ( D `  F ) ) )
97eqeq2d 2307 . . . . . 6  |-  ( f  =  F  ->  (
( C `  h
)  =  ( D `
 f )  <->  ( C `  h )  =  ( D `  F ) ) )
108, 93anbi23d 1255 . . . . 5  |-  ( f  =  F  ->  (
( ( D `  g )  =  ( D `  h )  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) )  <->  ( ( D `
 g )  =  ( D `  h
)  /\  ( C `  g )  =  ( D `  F )  /\  ( C `  h )  =  ( D `  F ) ) ) )
11 oveq1 5881 . . . . . . 7  |-  ( f  =  F  ->  (
f R g )  =  ( F R g ) )
12 oveq1 5881 . . . . . . 7  |-  ( f  =  F  ->  (
f R h )  =  ( F R h ) )
1311, 12eqeq12d 2310 . . . . . 6  |-  ( f  =  F  ->  (
( f R g )  =  ( f R h )  <->  ( F R g )  =  ( F R h ) ) )
1413imbi1d 308 . . . . 5  |-  ( f  =  F  ->  (
( ( f R g )  =  ( f R h )  ->  g  =  h )  <->  ( ( F R g )  =  ( F R h )  ->  g  =  h ) ) )
1510, 14imbi12d 311 . . . 4  |-  ( f  =  F  ->  (
( ( ( D `
 g )  =  ( D `  h
)  /\  ( C `  g )  =  ( D `  f )  /\  ( C `  h )  =  ( D `  f ) )  ->  ( (
f R g )  =  ( f R h )  ->  g  =  h ) )  <->  ( (
( D `  g
)  =  ( D `
 h )  /\  ( C `  g )  =  ( D `  F )  /\  ( C `  h )  =  ( D `  F ) )  -> 
( ( F R g )  =  ( F R h )  ->  g  =  h ) ) ) )
16152ralbidv 2598 . . 3  |-  ( f  =  F  ->  ( 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 ) )  <->  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 ) ) ) )
1716elrab 2936 . 2  |-  ( F  e.  { 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 ) ) }  <->  ( 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 ) ) ) )
186, 17syl6bb 252 1  |-  ( T  e.  Cat OLD  ->  ( F  e.  ( 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   dom cdm 4705   ` cfv 5271  (class class class)co 5874   dom_cdom_ 25815   cod_ccod_ 25816   o_co_ 25818    Cat
OLD ccatOLD 25855   MonoOLD cmonOLD 25907
This theorem is referenced by:  ismonb1  25914  ismonc  25917
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-monOLD 25909
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