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Mirrors > Home > MPE Home > Th. List > 0subcat | Structured version Visualization version GIF version |
Description: For any category 𝐶, the empty set is a (full) subcategory of 𝐶, see example 4.3(1.a) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.) |
Ref | Expression |
---|---|
0subcat | ⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ssc 17101 | . 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶)) | |
2 | ral0 4455 | . . 3 ⊢ ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥∅𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥∅𝑦)∀𝑔 ∈ (𝑦∅𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥∅𝑧)) | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥∅𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥∅𝑦)∀𝑔 ∈ (𝑦∅𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥∅𝑧))) |
4 | eqid 2821 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
5 | eqid 2821 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
6 | eqid 2821 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
7 | id 22 | . . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
8 | f0 6554 | . . . . . 6 ⊢ ∅:∅⟶∅ | |
9 | ffn 6508 | . . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ∅ Fn ∅ |
11 | 0xp 5643 | . . . . . 6 ⊢ (∅ × ∅) = ∅ | |
12 | 11 | fneq2i 6445 | . . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅) |
13 | 10, 12 | mpbir 233 | . . . 4 ⊢ ∅ Fn (∅ × ∅) |
14 | 13 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅)) |
15 | 4, 5, 6, 7, 14 | issubc2 17100 | . 2 ⊢ (𝐶 ∈ Cat → (∅ ∈ (Subcat‘𝐶) ↔ (∅ ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥∅𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥∅𝑦)∀𝑔 ∈ (𝑦∅𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥∅𝑧))))) |
16 | 1, 3, 15 | mpbir2and 711 | 1 ⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ∅c0 4290 〈cop 4566 class class class wbr 5058 × cxp 5547 Fn wfn 6344 ⟶wf 6345 ‘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-1st 7683 df-2nd 7684 df-pm 8403 df-ixp 8456 df-homf 16935 df-ssc 17074 df-subc 17076 |
This theorem is referenced by: (None) |
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