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Theorem ismonb2 25812
Description: A monomorphism is a left-cancelable morphism. (Contributed by FL, 2-Jan-2008.)
Hypotheses
Ref Expression
ismonb2.1  |-  M  =  dom  ( dom_ `  T
)
ismonb2.2  |-  D  =  ( dom_ `  T
)
ismonb2.3  |-  C  =  ( cod_ `  T
)
ismonb2.4  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
ismonb2  |-  ( ( T  e.  Cat OLD  /\  ( F  e.  M  /\  G  e.  M  /\  J  e.  M
) )  ->  ( F  e.  ( MonoOLD  `  T
)  ->  ( (
( D `  G
)  =  ( D `
 J )  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  J )  =  ( D `  F ) )  -> 
( ( F R G )  =  ( F R J )  ->  G  =  J ) ) ) )

Proof of Theorem ismonb2
Dummy variables  g 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismonb2.1 . . . 4  |-  M  =  dom  ( dom_ `  T
)
2 ismonb2.2 . . . 4  |-  D  =  ( dom_ `  T
)
3 ismonb2.3 . . . 4  |-  C  =  ( cod_ `  T
)
4 ismonb2.4 . . . 4  |-  R  =  ( o_ `  T
)
51, 2, 3, 4ismonb1 25811 . . 3  |-  ( ( T  e.  Cat OLD  /\  F  e.  M )  ->  ( F  e.  ( MonoOLD  `
 T )  <->  A. g  e.  M  A. j  e.  M  ( (
( D `  g
)  =  ( D `
 j )  /\  ( C `  g )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) )  -> 
( ( F R g )  =  ( F R j )  ->  g  =  j ) ) ) )
653ad2antr1 1120 . 2  |-  ( ( T  e.  Cat OLD  /\  ( F  e.  M  /\  G  e.  M  /\  J  e.  M
) )  ->  ( F  e.  ( MonoOLD  `  T
)  <->  A. g  e.  M  A. j  e.  M  ( ( ( D `
 g )  =  ( D `  j
)  /\  ( C `  g )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) )  ->  ( ( F R g )  =  ( F R j )  ->  g  =  j ) ) ) )
7 3simpc 954 . . . 4  |-  ( ( F  e.  M  /\  G  e.  M  /\  J  e.  M )  ->  ( G  e.  M  /\  J  e.  M
) )
87adantl 452 . . 3  |-  ( ( T  e.  Cat OLD  /\  ( F  e.  M  /\  G  e.  M  /\  J  e.  M
) )  ->  ( G  e.  M  /\  J  e.  M )
)
9 fveq2 5525 . . . . . . 7  |-  ( g  =  G  ->  ( D `  g )  =  ( D `  G ) )
109eqeq1d 2291 . . . . . 6  |-  ( g  =  G  ->  (
( D `  g
)  =  ( D `
 j )  <->  ( D `  G )  =  ( D `  j ) ) )
11 fveq2 5525 . . . . . . 7  |-  ( g  =  G  ->  ( C `  g )  =  ( C `  G ) )
1211eqeq1d 2291 . . . . . 6  |-  ( g  =  G  ->  (
( C `  g
)  =  ( D `
 F )  <->  ( C `  G )  =  ( D `  F ) ) )
1310, 123anbi12d 1253 . . . . 5  |-  ( g  =  G  ->  (
( ( D `  g )  =  ( D `  j )  /\  ( C `  g )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) )  <->  ( ( D `
 G )  =  ( D `  j
)  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) ) ) )
14 oveq2 5866 . . . . . . 7  |-  ( g  =  G  ->  ( F R g )  =  ( F R G ) )
1514eqeq1d 2291 . . . . . 6  |-  ( g  =  G  ->  (
( F R g )  =  ( F R j )  <->  ( F R G )  =  ( F R j ) ) )
16 eqeq1 2289 . . . . . 6  |-  ( g  =  G  ->  (
g  =  j  <->  G  =  j ) )
1715, 16imbi12d 311 . . . . 5  |-  ( g  =  G  ->  (
( ( F R g )  =  ( F R j )  ->  g  =  j )  <->  ( ( F R G )  =  ( F R j )  ->  G  =  j ) ) )
1813, 17imbi12d 311 . . . 4  |-  ( g  =  G  ->  (
( ( ( D `
 g )  =  ( D `  j
)  /\  ( C `  g )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) )  ->  ( ( F R g )  =  ( F R j )  ->  g  =  j ) )  <->  ( (
( D `  G
)  =  ( D `
 j )  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) )  -> 
( ( F R G )  =  ( F R j )  ->  G  =  j ) ) ) )
19 fveq2 5525 . . . . . . 7  |-  ( j  =  J  ->  ( D `  j )  =  ( D `  J ) )
2019eqeq2d 2294 . . . . . 6  |-  ( j  =  J  ->  (
( D `  G
)  =  ( D `
 j )  <->  ( D `  G )  =  ( D `  J ) ) )
21 fveq2 5525 . . . . . . 7  |-  ( j  =  J  ->  ( C `  j )  =  ( C `  J ) )
2221eqeq1d 2291 . . . . . 6  |-  ( j  =  J  ->  (
( C `  j
)  =  ( D `
 F )  <->  ( C `  J )  =  ( D `  F ) ) )
2320, 223anbi13d 1254 . . . . 5  |-  ( j  =  J  ->  (
( ( D `  G )  =  ( D `  j )  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) )  <->  ( ( D `
 G )  =  ( D `  J
)  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  J )  =  ( D `  F ) ) ) )
24 oveq2 5866 . . . . . . 7  |-  ( j  =  J  ->  ( F R j )  =  ( F R J ) )
2524eqeq2d 2294 . . . . . 6  |-  ( j  =  J  ->  (
( F R G )  =  ( F R j )  <->  ( F R G )  =  ( F R J ) ) )
26 eqeq2 2292 . . . . . 6  |-  ( j  =  J  ->  ( G  =  j  <->  G  =  J ) )
2725, 26imbi12d 311 . . . . 5  |-  ( j  =  J  ->  (
( ( F R G )  =  ( F R j )  ->  G  =  j )  <->  ( ( F R G )  =  ( F R J )  ->  G  =  J ) ) )
2823, 27imbi12d 311 . . . 4  |-  ( j  =  J  ->  (
( ( ( D `
 G )  =  ( D `  j
)  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) )  ->  ( ( F R G )  =  ( F R j )  ->  G  =  j ) )  <->  ( (
( D `  G
)  =  ( D `
 J )  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  J )  =  ( D `  F ) )  -> 
( ( F R G )  =  ( F R J )  ->  G  =  J ) ) ) )
2918, 28rspc2v 2890 . . 3  |-  ( ( G  e.  M  /\  J  e.  M )  ->  ( A. g  e.  M  A. j  e.  M  ( ( ( D `  g )  =  ( D `  j )  /\  ( C `  g )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) )  -> 
( ( F R g )  =  ( F R j )  ->  g  =  j ) )  ->  (
( ( D `  G )  =  ( D `  J )  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  J )  =  ( D `  F ) )  ->  ( ( F R G )  =  ( F R J )  ->  G  =  J ) ) ) )
308, 29syl 15 . 2  |-  ( ( T  e.  Cat OLD  /\  ( F  e.  M  /\  G  e.  M  /\  J  e.  M
) )  ->  ( A. g  e.  M  A. j  e.  M  ( ( ( D `
 g )  =  ( D `  j
)  /\  ( C `  g )  =  ( D `  F )  /\  ( C `  j )  =  ( D `  F ) )  ->  ( ( F R g )  =  ( F R j )  ->  g  =  j ) )  -> 
( ( ( D `
 G )  =  ( D `  J
)  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  J )  =  ( D `  F ) )  ->  ( ( F R G )  =  ( F R J )  ->  G  =  J ) ) ) )
316, 30sylbid 206 1  |-  ( ( T  e.  Cat OLD  /\  ( F  e.  M  /\  G  e.  M  /\  J  e.  M
) )  ->  ( F  e.  ( MonoOLD  `  T
)  ->  ( (
( D `  G
)  =  ( D `
 J )  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  J )  =  ( D `  F ) )  -> 
( ( F R G )  =  ( F R J )  ->  G  =  J ) ) ) )
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 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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