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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 2821 | . . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
4 | eqid 2821 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | subcrcl 17080 | . . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) | |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | eqidd 2822 | . . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
8 | 2, 3, 4, 6, 7 | issubc 17099 | . . 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 234 | . 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 497 | 1 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 〈cop 4566 class class class wbr 5058 dom cdm 5549 ‘cfv 6349 (class class class)co 7150 compcco 16571 Catccat 16929 Idccid 16930 Homf chomf 16931 ⊆cat cssc 17071 Subcatcsubc 17073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-pm 8403 df-ixp 8456 df-ssc 17074 df-subc 17076 |
This theorem is referenced by: subcfn 17105 subcss1 17106 subcss2 17107 issubc3 17113 subsubc 17117 |
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