Theorem subcss2 17893

Description: The morphisms of a subcategory are a subset of the morphisms 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))
subcss2.h	⊢ 𝐻 = (Hom ‘𝐶)
subcss2.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
subcss2.y	⊢ (𝜑 → 𝑌 ∈ 𝑆)

Assertion

Ref	Expression
subcss2	⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))

Proof of Theorem subcss2

Step	Hyp	Ref	Expression
1		subcss1.2	. . 3 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
2		subcss1.1	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
3		eqid 2734	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4	2, 3	subcssc 17890	. . 3 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶))
5		subcss2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
6		subcss2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
7	1, 4, 5, 6	ssc2 17869	. 2 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf ‘𝐶)𝑌))
8		eqid 2734	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		subcss2.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
10	2, 1, 8	subcss1 17892	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶))
11	10, 5	sseldd 3995	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12	10, 6	sseldd 3995	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
13	3, 8, 9, 11, 12	homfval 17736	. 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌))
14	7, 13	sseqtrd 4035	1 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2105   ⊆ wss 3962   × cxp 5686   Fn wfn 6557  ‘cfv 6562  (class class class)co 7430  Basecbs 17244  Hom chom 17308  Hom_f 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:  subccatid  17896  funcres  17946  funcres2b  17947  subthinc  48839