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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 2651 | . . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
4 | eqid 2651 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | subcrcl 16523 | . . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) | |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | eqidd 2652 | . . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
8 | 2, 3, 4, 6, 7 | issubc 16542 | . . 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 222 | . 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 1523 ∈ wcel 2030 ∀wral 2941 〈cop 4216 class class class wbr 4685 dom cdm 5143 ‘cfv 5926 (class class class)co 6690 compcco 16000 Catccat 16372 Idccid 16373 Homf chomf 16374 ⊆cat cssc 16514 Subcatcsubc 16516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-pm 7902 df-ixp 7951 df-ssc 16517 df-subc 16519 |
This theorem is referenced by: subcfn 16548 subcss1 16549 subcss2 16550 issubc3 16556 subsubc 16560 |
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