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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 17109 | . 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶)) | |
2 | ral0 4458 | . . 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 2823 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
5 | eqid 2823 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
6 | eqid 2823 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
7 | id 22 | . . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
8 | f0 6562 | . . . . . 6 ⊢ ∅:∅⟶∅ | |
9 | ffn 6516 | . . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ∅ Fn ∅ |
11 | 0xp 5651 | . . . . . 6 ⊢ (∅ × ∅) = ∅ | |
12 | 11 | fneq2i 6453 | . . . . 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 17108 | . 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 2114 ∀wral 3140 ∅c0 4293 〈cop 4575 class class class wbr 5068 × cxp 5555 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 compcco 16579 Catccat 16937 Idccid 16938 Homf chomf 16939 ⊆cat cssc 17079 Subcatcsubc 17081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-pm 8411 df-ixp 8464 df-homf 16943 df-ssc 17082 df-subc 17084 |
This theorem is referenced by: (None) |
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