Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  immon Unicode version

Theorem immon 25818
Description: A morphism identity is a monomorphism. JFM CAT1 th. 64. (Contributed by FL, 5-May-2008.)
Hypotheses
Ref Expression
immon.1  |-  O  =  dom  ( id_ `  T
)
immon.2  |-  J  =  ( id_ `  T
)
Assertion
Ref Expression
immon  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( J `  A )  e.  ( MonoOLD  `  T ) )

Proof of Theorem immon
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  T  e.  Cat OLD  )
2 simpr 447 . . . 4  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  A  e.  O
)
32, 2jca 518 . . 3  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( A  e.  O  /\  A  e.  O ) )
4 immon.1 . . . . 5  |-  O  =  dom  ( id_ `  T
)
5 immon.2 . . . . 5  |-  J  =  ( id_ `  T
)
6 eqid 2283 . . . . 5  |-  ( hom `  T )  =  ( hom `  T )
74, 5, 6homib 25796 . . . 4  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( J `  A )  e.  ( ( hom `  T
) `  <. A ,  A >. ) )
87, 7jca 518 . . 3  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( J `
 A )  e.  ( ( hom `  T
) `  <. A ,  A >. )  /\  ( J `  A )  e.  ( ( hom `  T
) `  <. A ,  A >. ) ) )
91, 3, 83jca 1132 . 2  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( T  e. 
Cat OLD  /\  ( A  e.  O  /\  A  e.  O )  /\  ( ( J `  A )  e.  ( ( hom `  T
) `  <. A ,  A >. )  /\  ( J `  A )  e.  ( ( hom `  T
) `  <. A ,  A >. ) ) ) )
10 eqid 2283 . . 3  |-  ( o_
`  T )  =  ( o_ `  T
)
114, 5, 10iri 25800 . 2  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( J `
 A ) ( o_ `  T ) ( J `  A
) )  =  ( J `  A ) )
124, 6, 10, 5idmon 25817 . 2  |-  ( ( T  e.  Cat OLD  /\  ( A  e.  O  /\  A  e.  O
)  /\  ( ( J `  A )  e.  ( ( hom `  T
) `  <. A ,  A >. )  /\  ( J `  A )  e.  ( ( hom `  T
) `  <. A ,  A >. ) ) )  ->  ( ( ( J `  A ) ( o_ `  T
) ( J `  A ) )  =  ( J `  A
)  ->  ( J `  A )  e.  ( MonoOLD  `  T ) ) )
139, 11, 12sylc 56 1  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( J `  A )  e.  ( MonoOLD  `  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   dom cdm 4689   ` cfv 5255  (class class class)co 5858   id_cid_ 25714   o_co_ 25715    Cat
OLD ccatOLD 25752   homchomOLD 25785   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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-alg 25716  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-ded 25735  df-catOLD 25753  df-homOLD 25786  df-monOLD 25806
  Copyright terms: Public domain W3C validator