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Mirrors > Home > MPE Home > Th. List > subcssc | Structured version Visualization version GIF version |
Description: An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
subcixp.1 | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
subcssc.h | ⊢ 𝐻 = (Homf ‘𝐶) |
Ref | Expression |
---|---|
subcssc | ⊢ (𝜑 → 𝐽 ⊆cat 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcixp.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
2 | subcssc.h | . . . 4 ⊢ 𝐻 = (Homf ‘𝐶) | |
3 | eqid 2610 | . . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
4 | eqid 2610 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | subcrcl 16299 | . . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) | |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | eqidd 2611 | . . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
8 | 2, 3, 4, 6, 7 | issubc 16318 | . . 3 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐽(((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ dom dom 𝐽∀𝑧 ∈ dom dom 𝐽∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))) |
9 | 1, 8 | mpbid 221 | . 2 ⊢ (𝜑 → (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐽(((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ dom dom 𝐽∀𝑧 ∈ dom dom 𝐽∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))) |
10 | 9 | simpld 474 | 1 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 〈cop 4131 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 compcco 15780 Catccat 16148 Idccid 16149 Homf chomf 16150 ⊆cat cssc 16290 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-pm 7747 df-ixp 7795 df-ssc 16293 df-subc 16295 |
This theorem is referenced by: subcfn 16324 subcss1 16325 subcss2 16326 issubc3 16332 subsubc 16336 |
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