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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 17826 | . 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶)) | |
2 | ral0 4514 | . . 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 2725 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
5 | eqid 2725 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
6 | eqid 2725 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
7 | id 22 | . . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
8 | f0 6778 | . . . . . 6 ⊢ ∅:∅⟶∅ | |
9 | ffn 6723 | . . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ∅ Fn ∅ |
11 | 0xp 5776 | . . . . . 6 ⊢ (∅ × ∅) = ∅ | |
12 | 11 | fneq2i 6653 | . . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅) |
13 | 10, 12 | mpbir 230 | . . . 4 ⊢ ∅ Fn (∅ × ∅) |
14 | 13 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅)) |
15 | 4, 5, 6, 7, 14 | issubc2 17825 | . 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 394 ∈ wcel 2098 ∀wral 3050 ∅c0 4322 〈cop 4636 class class class wbr 5149 × cxp 5676 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 compcco 17248 Catccat 17647 Idccid 17648 Homf chomf 17649 ⊆cat cssc 17793 Subcatcsubc 17795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-pm 8848 df-ixp 8917 df-homf 17653 df-ssc 17796 df-subc 17798 |
This theorem is referenced by: (None) |
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