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Theorem mrdmcd 25794
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 25764 . . . . 5  |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
2 dedalg 25743 . . . . 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 2287 . . . . . . . 8  |-  ( dom_ `  T )  =  D
76dmeqi 4880 . . . . . . 7  |-  dom  ( dom_ `  T )  =  dom  D
84, 7eqtri 2303 . . . . . 6  |-  M  =  dom  D
9 eqid 2283 . . . . . 6  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
10 eqid 2283 . . . . . 6  |-  ( id_ `  T )  =  ( id_ `  T )
118, 5, 9, 10doma 25728 . . . . 5  |-  ( T  e.  Alg  ->  D : M --> dom  ( id_ `  T ) )
12 ffvelrn 5663 . . . . . 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 25729 . . . . 5  |-  ( T  e.  Alg  ->  C : M --> dom  ( id_ `  T ) )
18 ffvelrn 5663 . . . . . 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 2283 . . . . 5  |-  ( D `
 F )  =  ( D `  F
)
24 eqid 2283 . . . . 5  |-  ( C `
 F )  =  ( C `  F
)
25 mrdmcd.2 . . . . . . . . 9  |-  H  =  ( hom `  T
)
269, 4, 5, 16, 25ishomd 25790 . . . . . . . 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 1623    e. wcel 1684   <.cop 3643   dom cdm 4689   -->wf 5251   ` cfv 5255    Alg calg 25711   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   Dedcded 25734    Cat
OLD ccatOLD 25752   homchomOLD 25785
This theorem is referenced by:  homib  25796
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
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