Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2818 | . . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
4 | eqid 2818 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | subcrcl 17074 | . . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) | |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | eqidd 2819 | . . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
8 | 2, 3, 4, 6, 7 | issubc 17093 | . . 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 233 | . 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 495 | 1 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 〈cop 4563 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 compcco 16565 Catccat 16923 Idccid 16924 Homf chomf 16925 ⊆cat cssc 17065 Subcatcsubc 17067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-pm 8398 df-ixp 8450 df-ssc 17068 df-subc 17070 |
This theorem is referenced by: subcfn 17099 subcss1 17100 subcss2 17101 issubc3 17107 subsubc 17111 |
Copyright terms: Public domain | W3C validator |