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Theorem subcss1 15190
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 2443 . . . 4  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
3 subcss1.b . . . 4  |-  B  =  ( Base `  C
)
42, 3homffn 15070 . . 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 15188 . 2  |-  ( ph  ->  J  C_cat  ( Hom f  `  C ) )
81, 5, 7ssc1 15172 1  |-  ( ph  ->  S  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804    C_ wss 3461    X. cxp 4987    Fn wfn 5573   ` cfv 5578   Basecbs 14614   Hom f chomf 15045  Subcatcsubc 15160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-pm 7425  df-ixp 7472  df-homf 15049  df-ssc 15161  df-subc 15163
This theorem is referenced by:  subcss2  15191  subccatid  15194  subsubc  15201  funcres  15244  funcres2b  15245  funcres2  15246  idfusubc  32516
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