Theorem subcss2 16888

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 2777	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4	2, 3	subcssc 16885	. . 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 16867	. 2 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf ‘𝐶)𝑌))
8		eqid 2777	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		subcss2.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
10	2, 1, 8	subcss1 16887	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶))
11	10, 5	sseldd 3821	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12	10, 6	sseldd 3821	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
13	3, 8, 9, 11, 12	homfval 16737	. 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌))
14	7, 13	sseqtrd 3859	1 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2106   ⊆ wss 3791   × cxp 5353   Fn wfn 6130  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  Hom chom 16349  Hom_f chomf 16712  Subcatcsubc 16854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-pm 8143  df-ixp 8195  df-homf 16716  df-ssc 16855  df-subc 16857

This theorem is referenced by:  subccatid  16891  funcres  16941  funcres2b  16942