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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 16718 | . 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶)) | |
2 | ral0 4220 | . . 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 2760 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
5 | eqid 2760 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
6 | eqid 2760 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
7 | id 22 | . . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
8 | f0 6247 | . . . . . 6 ⊢ ∅:∅⟶∅ | |
9 | ffn 6206 | . . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ∅ Fn ∅ |
11 | 0xp 5356 | . . . . . 6 ⊢ (∅ × ∅) = ∅ | |
12 | 11 | fneq2i 6147 | . . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅) |
13 | 10, 12 | mpbir 221 | . . . 4 ⊢ ∅ Fn (∅ × ∅) |
14 | 13 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅)) |
15 | 4, 5, 6, 7, 14 | issubc2 16717 | . 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 995 | 1 ⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2139 ∀wral 3050 ∅c0 4058 〈cop 4327 class class class wbr 4804 × cxp 5264 Fn wfn 6044 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 compcco 16175 Catccat 16546 Idccid 16547 Homf chomf 16548 ⊆cat cssc 16688 Subcatcsubc 16690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-1st 7334 df-2nd 7335 df-pm 8028 df-ixp 8077 df-homf 16552 df-ssc 16691 df-subc 16693 |
This theorem is referenced by: (None) |
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