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Theorem subcss2 13717
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 2283 . . . 4  |-  (  Homf  `  C )  =  (  Homf 
`  C )
42, 3subcssc 13714 . . 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 13699 . 2  |-  ( ph  ->  ( X J Y )  C_  ( X
(  Homf 
`  C ) Y ) )
8 eqid 2283 . . 3  |-  ( Base `  C )  =  (
Base `  C )
9 subcss2.h . . 3  |-  H  =  (  Hom  `  C
)
102, 1, 8subcss1 13716 . . . 4  |-  ( ph  ->  S  C_  ( Base `  C ) )
1110, 5sseldd 3181 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
1210, 6sseldd 3181 . . 3  |-  ( ph  ->  Y  e.  ( Base `  C ) )
133, 8, 9, 11, 12homfval 13595 . 2  |-  ( ph  ->  ( X (  Homf  `  C ) Y )  =  ( X H Y ) )
147, 13sseqtrd 3214 1  |-  ( ph  ->  ( X J Y )  C_  ( X H Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219    Homf chomf 13568  Subcatcsubc 13686
This theorem is referenced by:  subccatid  13720  funcres  13770  funcres2b  13771
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-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-pm 6775  df-ixp 6818  df-homf 13572  df-ssc 13687  df-subc 13689
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