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Theorem subcss1 17892
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 2734 . . . 4 (Homf𝐶) = (Homf𝐶)
3 subcss1.b . . . 4 𝐵 = (Base‘𝐶)
42, 3homffn 17737 . . 3 (Homf𝐶) Fn (𝐵 × 𝐵)
54a1i 11 . 2 (𝜑 → (Homf𝐶) Fn (𝐵 × 𝐵))
6 subcss1.1 . . 3 (𝜑𝐽 ∈ (Subcat‘𝐶))
76, 2subcssc 17890 . 2 (𝜑𝐽cat (Homf𝐶))
81, 5, 7ssc1 17868 1 (𝜑𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  wss 3962   × cxp 5686   Fn wfn 6557  cfv 6562  Basecbs 17244  Homf chomf 17710  Subcatcsubc 17856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-pm 8867  df-ixp 8936  df-homf 17714  df-ssc 17857  df-subc 17859
This theorem is referenced by:  subcss2  17893  subccatid  17896  subsubc  17903  funcres  17946  funcres2b  17947  funcres2  17948  idfusubc  17950  subthinc  48839
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