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Theorem mrdmcd 25897
Description: A morphism belongs to the homset between its domain and its codomain. JFM CAT1 th. 22. (Contributed by FL, 1-Nov-2007.)
Hypotheses
Ref Expression
mrdmcd.1  |-  M  =  dom  ( dom_ `  T
)
mrdmcd.2  |-  H  =  ( hom `  T
)
mrdmcd.3  |-  D  =  ( dom_ `  T
)
mrdmcd.4  |-  C  =  ( cod_ `  T
)
Assertion
Ref Expression
mrdmcd  |-  ( T  e.  Cat OLD  ->  ( F  e.  M  ->  F  e.  ( H `  <. ( D `  F ) ,  ( C `  F )
>. ) ) )

Proof of Theorem mrdmcd
StepHypRef Expression
1 catded 25867 . . . . 5  |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
2 dedalg 25846 . . . . 5  |-  ( T  e.  Ded  ->  T  e.  Alg  )
31, 2syl 15 . . . 4  |-  ( T  e.  Cat OLD  ->  T  e.  Alg  )
4 mrdmcd.1 . . . . . . 7  |-  M  =  dom  ( dom_ `  T
)
5 mrdmcd.3 . . . . . . . . 9  |-  D  =  ( dom_ `  T
)
65eqcomi 2300 . . . . . . . 8  |-  ( dom_ `  T )  =  D
76dmeqi 4896 . . . . . . 7  |-  dom  ( dom_ `  T )  =  dom  D
84, 7eqtri 2316 . . . . . 6  |-  M  =  dom  D
9 eqid 2296 . . . . . 6  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
10 eqid 2296 . . . . . 6  |-  ( id_ `  T )  =  ( id_ `  T )
118, 5, 9, 10doma 25831 . . . . 5  |-  ( T  e.  Alg  ->  D : M --> dom  ( id_ `  T ) )
12 ffvelrn 5679 . . . . . 6  |-  ( ( D : M --> dom  ( id_ `  T )  /\  F  e.  M )  ->  ( D `  F
)  e.  dom  ( id_ `  T ) )
1312ex 423 . . . . 5  |-  ( D : M --> dom  ( id_ `  T )  -> 
( F  e.  M  ->  ( D `  F
)  e.  dom  ( id_ `  T ) ) )
1411, 13syl 15 . . . 4  |-  ( T  e.  Alg  ->  ( F  e.  M  ->  ( D `  F )  e.  dom  ( id_ `  T ) ) )
153, 14syl 15 . . 3  |-  ( T  e.  Cat OLD  ->  ( F  e.  M  -> 
( D `  F
)  e.  dom  ( id_ `  T ) ) )
16 mrdmcd.4 . . . . . 6  |-  C  =  ( cod_ `  T
)
178, 5, 9, 10, 16coda 25832 . . . . 5  |-  ( T  e.  Alg  ->  C : M --> dom  ( id_ `  T ) )
18 ffvelrn 5679 . . . . . 6  |-  ( ( C : M --> dom  ( id_ `  T )  /\  F  e.  M )  ->  ( C `  F
)  e.  dom  ( id_ `  T ) )
1918ex 423 . . . . 5  |-  ( C : M --> dom  ( id_ `  T )  -> 
( F  e.  M  ->  ( C `  F
)  e.  dom  ( id_ `  T ) ) )
2017, 19syl 15 . . . 4  |-  ( T  e.  Alg  ->  ( F  e.  M  ->  ( C `  F )  e.  dom  ( id_ `  T ) ) )
213, 20syl 15 . . 3  |-  ( T  e.  Cat OLD  ->  ( F  e.  M  -> 
( C `  F
)  e.  dom  ( id_ `  T ) ) )
2215, 21jcad 519 . 2  |-  ( T  e.  Cat OLD  ->  ( F  e.  M  -> 
( ( D `  F )  e.  dom  ( id_ `  T )  /\  ( C `  F )  e.  dom  ( id_ `  T ) ) ) )
23 eqid 2296 . . . . 5  |-  ( D `
 F )  =  ( D `  F
)
24 eqid 2296 . . . . 5  |-  ( C `
 F )  =  ( C `  F
)
25 mrdmcd.2 . . . . . . . . 9  |-  H  =  ( hom `  T
)
269, 4, 5, 16, 25ishomd 25893 . . . . . . . 8  |-  ( ( T  e.  Cat OLD  /\  ( D `  F
)  e.  dom  ( id_ `  T )  /\  ( C `  F )  e.  dom  ( id_ `  T ) )  -> 
( F  e.  ( H `  <. ( D `  F ) ,  ( C `  F ) >. )  <->  ( F  e.  M  /\  ( D `  F )  =  ( D `  F )  /\  ( C `  F )  =  ( C `  F ) ) ) )
2726biimprcd 216 . . . . . . 7  |-  ( ( F  e.  M  /\  ( D `  F )  =  ( D `  F )  /\  ( C `  F )  =  ( C `  F ) )  -> 
( ( T  e. 
Cat OLD  /\  ( D `  F )  e.  dom  ( id_ `  T
)  /\  ( C `  F )  e.  dom  ( id_ `  T ) )  ->  F  e.  ( H `  <. ( D `  F ) ,  ( C `  F ) >. )
) )
28273expib 1154 . . . . . 6  |-  ( F  e.  M  ->  (
( ( D `  F )  =  ( D `  F )  /\  ( C `  F )  =  ( C `  F ) )  ->  ( ( T  e.  Cat OLD  /\  ( D `  F )  e.  dom  ( id_ `  T )  /\  ( C `  F )  e.  dom  ( id_ `  T
) )  ->  F  e.  ( H `  <. ( D `  F ) ,  ( C `  F ) >. )
) ) )
2928com3l 75 . . . . 5  |-  ( ( ( D `  F
)  =  ( D `
 F )  /\  ( C `  F )  =  ( C `  F ) )  -> 
( ( T  e. 
Cat OLD  /\  ( D `  F )  e.  dom  ( id_ `  T
)  /\  ( C `  F )  e.  dom  ( id_ `  T ) )  ->  ( F  e.  M  ->  F  e.  ( H `  <. ( D `  F ) ,  ( C `  F ) >. )
) ) )
3023, 24, 29mp2an 653 . . . 4  |-  ( ( T  e.  Cat OLD  /\  ( D `  F
)  e.  dom  ( id_ `  T )  /\  ( C `  F )  e.  dom  ( id_ `  T ) )  -> 
( F  e.  M  ->  F  e.  ( H `
 <. ( D `  F ) ,  ( C `  F )
>. ) ) )
31303expib 1154 . . 3  |-  ( T  e.  Cat OLD  ->  ( ( ( D `  F )  e.  dom  ( id_ `  T )  /\  ( C `  F )  e.  dom  ( id_ `  T ) )  ->  ( F  e.  M  ->  F  e.  ( H `  <. ( D `  F ) ,  ( C `  F ) >. )
) ) )
3231com23 72 . 2  |-  ( T  e.  Cat OLD  ->  ( F  e.  M  -> 
( ( ( D `
 F )  e. 
dom  ( id_ `  T
)  /\  ( C `  F )  e.  dom  ( id_ `  T ) )  ->  F  e.  ( H `  <. ( D `  F ) ,  ( C `  F ) >. )
) ) )
3322, 32mpdd 36 1  |-  ( T  e.  Cat OLD  ->  ( F  e.  M  ->  F  e.  ( H `  <. ( D `  F ) ,  ( C `  F )
>. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656   dom cdm 4705   -->wf 5267   ` cfv 5271    Alg calg 25814   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   Dedcded 25837    Cat
OLD ccatOLD 25855   homchomOLD 25888
This theorem is referenced by:  homib  25899
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
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