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Theorem subcss1 15086
Description: The objects of a subcategory are a subset of the objects 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 ) )
subcss1.b  |-  B  =  ( Base `  C
)
Assertion
Ref Expression
subcss1  |-  ( ph  ->  S  C_  B )

Proof of Theorem subcss1
StepHypRef Expression
1 subcss1.2 . 2  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
2 eqid 2467 . . . 4  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
3 subcss1.b . . . 4  |-  B  =  ( Base `  C
)
42, 3homffn 14966 . . 3  |-  ( Hom f  `  C )  Fn  ( B  X.  B )
54a1i 11 . 2  |-  ( ph  ->  ( Hom f  `  C )  Fn  ( B  X.  B
) )
6 subcss1.1 . . 3  |-  ( ph  ->  J  e.  (Subcat `  C ) )
76, 2subcssc 15084 . 2  |-  ( ph  ->  J  C_cat  ( Hom f  `  C ) )
81, 5, 7ssc1 15068 1  |-  ( ph  ->  S  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    C_ wss 3481    X. cxp 5003    Fn wfn 5589   ` cfv 5594   Basecbs 14507   Hom f chomf 14938  Subcatcsubc 15056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-pm 7435  df-ixp 7482  df-homf 14942  df-ssc 15057  df-subc 15059
This theorem is referenced by:  subcss2  15087  subccatid  15090  subsubc  15097  funcres  15140  funcres2b  15141  funcres2  15142
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