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Theorem ismonb1 25811
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
ismonb1  |-  ( ( T  e.  Cat OLD  /\  F  e.  M )  ->  ( F  e.  ( MonoOLD  `
 T )  <->  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 ismonb1
StepHypRef Expression
1 ismonb.1 . . 3  |-  M  =  dom  ( dom_ `  T
)
2 ismonb.2 . . 3  |-  D  =  ( dom_ `  T
)
3 ismonb.3 . . 3  |-  C  =  ( cod_ `  T
)
4 ismonb.4 . . 3  |-  R  =  ( o_ `  T
)
51, 2, 3, 4ismonb 25810 . 2  |-  ( 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 ) ) ) ) )
65baibd 875 1  |-  ( ( T  e.  Cat OLD  /\  F  e.  M )  ->  ( F  e.  ( MonoOLD  `
 T )  <->  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 1623    e. wcel 1684   A.wral 2543   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:  ismonb2  25812
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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