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Theorem subcss1 15345
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 2400 . . . 4  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
3 subcss1.b . . . 4  |-  B  =  ( Base `  C
)
42, 3homffn 15196 . . 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 15343 . 2  |-  ( ph  ->  J  C_cat  ( Hom f  `  C ) )
81, 5, 7ssc1 15324 1  |-  ( ph  ->  S  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840    C_ wss 3411    X. cxp 4938    Fn wfn 5518   ` cfv 5523   Basecbs 14731   Hom f chomf 15170  Subcatcsubc 15312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-pm 7378  df-ixp 7426  df-homf 15174  df-ssc 15313  df-subc 15315
This theorem is referenced by:  subcss2  15346  subccatid  15349  subsubc  15356  funcres  15399  funcres2b  15400  funcres2  15401  idfusubc  38116
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