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Theorem subcss1 13716
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 2283 . . . 4  |-  (  Homf  `  C )  =  (  Homf 
`  C )
3 subcss1.b . . . 4  |-  B  =  ( Base `  C
)
42, 3homffn 13596 . . 3  |-  (  Homf  `  C )  Fn  ( B  X.  B )
54a1i 10 . 2  |-  ( ph  ->  (  Homf 
`  C )  Fn  ( B  X.  B
) )
6 subcss1.1 . . 3  |-  ( ph  ->  J  e.  (Subcat `  C ) )
76, 2subcssc 13714 . 2  |-  ( ph  ->  J  C_cat  (  Homf 
`  C ) )
81, 5, 7ssc1 13698 1  |-  ( ph  ->  S  C_  B )
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   Basecbs 13148    Homf chomf 13568  Subcatcsubc 13686
This theorem is referenced by:  subcss2  13717  subccatid  13720  subsubc  13727  funcres  13770  funcres2b  13771  funcres2  13772
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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