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Theorem subcss1 17114
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 2824 . . . 4 (Homf𝐶) = (Homf𝐶)
3 subcss1.b . . . 4 𝐵 = (Base‘𝐶)
42, 3homffn 16965 . . 3 (Homf𝐶) Fn (𝐵 × 𝐵)
54a1i 11 . 2 (𝜑 → (Homf𝐶) Fn (𝐵 × 𝐵))
6 subcss1.1 . . 3 (𝜑𝐽 ∈ (Subcat‘𝐶))
76, 2subcssc 17112 . 2 (𝜑𝐽cat (Homf𝐶))
81, 5, 7ssc1 17093 1 (𝜑𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wss 3919   × cxp 5541   Fn wfn 6340  cfv 6345  Basecbs 16485  Homf chomf 16939  Subcatcsubc 17081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-pm 8407  df-ixp 8460  df-homf 16943  df-ssc 17082  df-subc 17084
This theorem is referenced by:  subcss2  17115  subccatid  17118  subsubc  17125  funcres  17168  funcres2b  17169  funcres2  17170  idfusubc  44443
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