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Mirrors > Home > MPE Home > Th. List > subcss2 | Structured version Visualization version GIF version |
Description: The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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) |
Ref | Expression |
---|---|
subcss2 | ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcss1.2 | . . 3 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) | |
2 | subcss1.1 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
3 | eqid 2610 | . . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
4 | 2, 3 | subcssc 16323 | . . 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 16305 | . 2 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf ‘𝐶)𝑌)) |
8 | eqid 2610 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | subcss2.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
10 | 2, 1, 8 | subcss1 16325 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶)) |
11 | 10, 5 | sseldd 3569 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
12 | 10, 6 | sseldd 3569 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
13 | 3, 8, 9, 11, 12 | homfval 16175 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌)) |
14 | 7, 13 | sseqtrd 3604 | 1 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 × cxp 5036 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Hom chom 15779 Homf chomf 16150 Subcatcsubc 16292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-pm 7747 df-ixp 7795 df-homf 16154 df-ssc 16293 df-subc 16295 |
This theorem is referenced by: subccatid 16329 funcres 16379 funcres2b 16380 |
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