Theorem subcss2 17113

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 2821	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4	2, 3	subcssc 17110	. . 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 17092	. 2 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf ‘𝐶)𝑌))
8		eqid 2821	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		subcss2.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
10	2, 1, 8	subcss1 17112	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶))
11	10, 5	sseldd 3968	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12	10, 6	sseldd 3968	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
13	3, 8, 9, 11, 12	homfval 16962	. 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌))
14	7, 13	sseqtrd 4007	1 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936   × cxp 5553   Fn wfn 6350  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  Hom chom 16576  Hom_f chomf 16937  Subcatcsubc 17079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-pm 8409  df-ixp 8462  df-homf 16941  df-ssc 17080  df-subc 17082

This theorem is referenced by:  subccatid  17116  funcres  17166  funcres2b  17167