Theorem subcss1 17887

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	⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
subcss1.2	⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
subcss1.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
subcss1	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Proof of Theorem subcss1

Step	Hyp	Ref	Expression
1		subcss1.2	. 2 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
2		eqid 2737	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
3		subcss1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	homffn 17736	. . 3 ⊢ (Homf ‘𝐶) Fn (𝐵 × 𝐵)
5	4	a1i 11	. 2 ⊢ (𝜑 → (Homf ‘𝐶) Fn (𝐵 × 𝐵))
6		subcss1.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
7	6, 2	subcssc 17885	. 2 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶))
8	1, 5, 7	ssc1 17865	1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951   × cxp 5683   Fn wfn 6556  ‘cfv 6561  Basecbs 17247  Hom_f chomf 17709  Subcatcsubc 17853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-pm 8869  df-ixp 8938  df-homf 17713  df-ssc 17854  df-subc 17856

This theorem is referenced by:  subcss2  17888  subccatid  17891  subsubc  17898  funcres  17941  funcres2b  17942  funcres2  17943  idfusubc  17945  subthinc  49092