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Theorem immon 25921
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 2296 . . . . 5  |-  ( hom `  T )  =  ( hom `  T )
74, 5, 6homib 25899 . . . 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 2296 . . 3  |-  ( o_
`  T )  =  ( o_ `  T
)
114, 5, 10iri 25903 . 2  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( J `
 A ) ( o_ `  T ) ( J `  A
) )  =  ( J `  A ) )
124, 6, 10, 5idmon 25920 . 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 1632    e. wcel 1696   <.cop 3656   dom cdm 4705   ` cfv 5271  (class class class)co 5874   id_cid_ 25817   o_co_ 25818    Cat
OLD ccatOLD 25855   homchomOLD 25888   MonoOLD cmonOLD 25907
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-alg 25819  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-ded 25838  df-catOLD 25856  df-homOLD 25889  df-monOLD 25909
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