Theorem subcss2 17779

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 2737	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4	2, 3	subcssc 17776	. . 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 17758	. 2 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf ‘𝐶)𝑌))
8		eqid 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		subcss2.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
10	2, 1, 8	subcss1 17778	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶))
11	10, 5	sseldd 3936	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12	10, 6	sseldd 3936	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
13	3, 8, 9, 11, 12	homfval 17627	. 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌))
14	7, 13	sseqtrd 3972	1 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903   × cxp 5630   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  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:  subccatid  17782  funcres  17832  funcres2b  17833  subthinc  49796