Theorem subcss2 17805

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 2729	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4	2, 3	subcssc 17802	. . 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 17784	. 2 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf ‘𝐶)𝑌))
8		eqid 2729	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		subcss2.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
10	2, 1, 8	subcss1 17804	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶))
11	10, 5	sseldd 3947	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12	10, 6	sseldd 3947	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
13	3, 8, 9, 11, 12	homfval 17653	. 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌))
14	7, 13	sseqtrd 3983	1 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914   × cxp 5636   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  Hom_f chomf 17627  Subcatcsubc 17771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-pm 8802  df-ixp 8871  df-homf 17631  df-ssc 17772  df-subc 17774

This theorem is referenced by:  subccatid  17808  funcres  17858  funcres2b  17859  subthinc  49432