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Theorem subcss2 13969
Description: The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcss1.1  |-  ( ph  ->  J  e.  (Subcat `  C ) )
subcss1.2  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
subcss2.h  |-  H  =  (  Hom  `  C
)
subcss2.x  |-  ( ph  ->  X  e.  S )
subcss2.y  |-  ( ph  ->  Y  e.  S )
Assertion
Ref Expression
subcss2  |-  ( ph  ->  ( X J Y )  C_  ( X H Y ) )

Proof of Theorem subcss2
StepHypRef Expression
1 subcss1.2 . . 3  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
2 subcss1.1 . . . 4  |-  ( ph  ->  J  e.  (Subcat `  C ) )
3 eqid 2389 . . . 4  |-  (  Homf  `  C )  =  (  Homf 
`  C )
42, 3subcssc 13966 . . 3  |-  ( ph  ->  J  C_cat  (  Homf 
`  C ) )
5 subcss2.x . . 3  |-  ( ph  ->  X  e.  S )
6 subcss2.y . . 3  |-  ( ph  ->  Y  e.  S )
71, 4, 5, 6ssc2 13951 . 2  |-  ( ph  ->  ( X J Y )  C_  ( X
(  Homf 
`  C ) Y ) )
8 eqid 2389 . . 3  |-  ( Base `  C )  =  (
Base `  C )
9 subcss2.h . . 3  |-  H  =  (  Hom  `  C
)
102, 1, 8subcss1 13968 . . . 4  |-  ( ph  ->  S  C_  ( Base `  C ) )
1110, 5sseldd 3294 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
1210, 6sseldd 3294 . . 3  |-  ( ph  ->  Y  e.  ( Base `  C ) )
133, 8, 9, 11, 12homfval 13847 . 2  |-  ( ph  ->  ( X (  Homf  `  C ) Y )  =  ( X H Y ) )
147, 13sseqtrd 3329 1  |-  ( ph  ->  ( X J Y )  C_  ( X H Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    C_ wss 3265    X. cxp 4818    Fn wfn 5391   ` cfv 5396  (class class class)co 6022   Basecbs 13398    Hom chom 13469    Homf chomf 13820  Subcatcsubc 13938
This theorem is referenced by:  subccatid  13972  funcres  14022  funcres2b  14023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-pm 6959  df-ixp 7002  df-homf 13824  df-ssc 13939  df-subc 13941
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