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Theorem subcss2 14030
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 2435 . . . 4  |-  (  Homf  `  C )  =  (  Homf 
`  C )
42, 3subcssc 14027 . . 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 14012 . 2  |-  ( ph  ->  ( X J Y )  C_  ( X
(  Homf 
`  C ) Y ) )
8 eqid 2435 . . 3  |-  ( Base `  C )  =  (
Base `  C )
9 subcss2.h . . 3  |-  H  =  (  Hom  `  C
)
102, 1, 8subcss1 14029 . . . 4  |-  ( ph  ->  S  C_  ( Base `  C ) )
1110, 5sseldd 3341 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
1210, 6sseldd 3341 . . 3  |-  ( ph  ->  Y  e.  ( Base `  C ) )
133, 8, 9, 11, 12homfval 13908 . 2  |-  ( ph  ->  ( X (  Homf  `  C ) Y )  =  ( X H Y ) )
147, 13sseqtrd 3376 1  |-  ( ph  ->  ( X J Y )  C_  ( X H Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3312    X. cxp 4868    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   Basecbs 13459    Hom chom 13530    Homf chomf 13881  Subcatcsubc 13999
This theorem is referenced by:  subccatid  14033  funcres  14083  funcres2b  14084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-pm 7013  df-ixp 7056  df-homf 13885  df-ssc 14000  df-subc 14002
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