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Theorem subcss1 17861
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 (𝜑𝐽 ∈ (Subcat‘𝐶))
subcss1.2 (𝜑𝐽 Fn (𝑆 × 𝑆))
subcss1.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
subcss1 (𝜑𝑆𝐵)

Proof of Theorem subcss1
StepHypRef Expression
1 subcss1.2 . 2 (𝜑𝐽 Fn (𝑆 × 𝑆))
2 eqid 2726 . . . 4 (Homf𝐶) = (Homf𝐶)
3 subcss1.b . . . 4 𝐵 = (Base‘𝐶)
42, 3homffn 17706 . . 3 (Homf𝐶) Fn (𝐵 × 𝐵)
54a1i 11 . 2 (𝜑 → (Homf𝐶) Fn (𝐵 × 𝐵))
6 subcss1.1 . . 3 (𝜑𝐽 ∈ (Subcat‘𝐶))
76, 2subcssc 17859 . 2 (𝜑𝐽cat (Homf𝐶))
81, 5, 7ssc1 17837 1 (𝜑𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wss 3947   × cxp 5680   Fn wfn 6549  cfv 6554  Basecbs 17213  Homf chomf 17679  Subcatcsubc 17825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-pm 8858  df-ixp 8927  df-homf 17683  df-ssc 17826  df-subc 17828
This theorem is referenced by:  subcss2  17862  subccatid  17865  subsubc  17872  funcres  17915  funcres2b  17916  funcres2  17917  idfusubc  17919  subthinc  48361
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