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Theorem morsubc 25855
Description: The morphisms of a subcategory are a subset of those of the supercategory. (Contributed by FL, 18-Sep-2009.)
Hypotheses
Ref Expression
morsubc.1  |-  M1  =  dom  ( dom_ `  T
)
morsubc.2  |-  M 2  =  dom  ( dom_ `  U
)
Assertion
Ref Expression
morsubc  |-  ( U  e.  (  SubCat  `  T
)  ->  M 2  C_  M1 )

Proof of Theorem morsubc
StepHypRef Expression
1 besubbeca 25848 . . . 4  |-  ( U  e.  (  SubCat  `  T
)  ->  T  e.  Cat OLD  )
2 eqid 2283 . . . . 5  |-  ( dom_ `  T )  =  (
dom_ `  T )
3 eqid 2283 . . . . 5  |-  ( cod_ `  T )  =  (
cod_ `  T )
4 eqid 2283 . . . . 5  |-  ( o_
`  T )  =  ( o_ `  T
)
5 eqid 2283 . . . . 5  |-  ( id_ `  T )  =  ( id_ `  T )
62, 3, 4, 5issubcata 25846 . . . 4  |-  ( T  e.  Cat OLD  ->  ( U  e.  (  SubCat  `  T )  <->  ( U  e.  Cat OLD  /\  (
( id_ `  U
)  C_  ( id_ `  T )  /\  (
( dom_ `  U )  C_  ( dom_ `  T
)  /\  ( cod_ `  U )  C_  ( cod_ `  T ) )  /\  ( o_ `  U )  C_  (
o_ `  T )
) ) ) )
71, 6syl 15 . . 3  |-  ( U  e.  (  SubCat  `  T
)  ->  ( U  e.  (  SubCat  `  T
)  <->  ( U  e. 
Cat OLD  /\  (
( id_ `  U
)  C_  ( id_ `  T )  /\  (
( dom_ `  U )  C_  ( dom_ `  T
)  /\  ( cod_ `  U )  C_  ( cod_ `  T ) )  /\  ( o_ `  U )  C_  (
o_ `  T )
) ) ) )
8 dmss 4878 . . . . . . 7  |-  ( (
dom_ `  U )  C_  ( dom_ `  T
)  ->  dom  ( dom_ `  U )  C_  dom  ( dom_ `  T )
)
9 morsubc.2 . . . . . . 7  |-  M 2  =  dom  ( dom_ `  U
)
10 morsubc.1 . . . . . . 7  |-  M1  =  dom  ( dom_ `  T
)
118, 9, 103sstr4g 3219 . . . . . 6  |-  ( (
dom_ `  U )  C_  ( dom_ `  T
)  ->  M 2  C_  M1 )
1211adantr 451 . . . . 5  |-  ( ( ( dom_ `  U
)  C_  ( dom_ `  T )  /\  ( cod_ `  U )  C_  ( cod_ `  T )
)  ->  M 2  C_  M1 )
13123ad2ant2 977 . . . 4  |-  ( ( ( id_ `  U
)  C_  ( id_ `  T )  /\  (
( dom_ `  U )  C_  ( dom_ `  T
)  /\  ( cod_ `  U )  C_  ( cod_ `  T ) )  /\  ( o_ `  U )  C_  (
o_ `  T )
)  ->  M 2  C_  M1 )
1413adantl 452 . . 3  |-  ( ( U  e.  Cat OLD  /\  ( ( id_ `  U
)  C_  ( id_ `  T )  /\  (
( dom_ `  U )  C_  ( dom_ `  T
)  /\  ( cod_ `  U )  C_  ( cod_ `  T ) )  /\  ( o_ `  U )  C_  (
o_ `  T )
) )  ->  M 2  C_  M1 )
157, 14syl6bi 219 . 2  |-  ( U  e.  (  SubCat  `  T
)  ->  ( U  e.  (  SubCat  `  T
)  ->  M 2  C_  M1 ) )
1615pm2.43i 43 1  |-  ( U  e.  (  SubCat  `  T
)  ->  M 2  C_  M1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   dom cdm 4689   ` cfv 5255   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715    Cat
OLD ccatOLD 25752    SubCat csubcat 25843
This theorem is referenced by:  idsubfun  25858  infemb  25859
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-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-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-pw 3627  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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-catOLD 25753  df-subcat 25844
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