Theorem subcss1 17778

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 17628	. . 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 17776	. 2 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶))
8	1, 5, 7	ssc1 17757	1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903   × cxp 5630   Fn wfn 6495  ‘cfv 6500  Basecbs 17148  Hom_f chomf 17601  Subcatcsubc 17745

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-pm 8778  df-ixp 8848  df-homf 17605  df-ssc 17746  df-subc 17748

This theorem is referenced by:  subcss2  17779  subccatid  17782  subsubc  17789  funcres  17832  funcres2b  17833  funcres2  17834  idfusubc  17836  subthinc  49796