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| 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 17882 | . 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶)) | |
| 2 | ral0 4513 | . . 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 2737 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
| 5 | eqid 2737 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
| 6 | eqid 2737 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
| 7 | id 22 | . . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
| 8 | f0 6789 | . . . . . 6 ⊢ ∅:∅⟶∅ | |
| 9 | ffn 6736 | . . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅) | |
| 10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ∅ Fn ∅ | 
| 11 | 0xp 5784 | . . . . . 6 ⊢ (∅ × ∅) = ∅ | |
| 12 | 11 | fneq2i 6666 | . . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅) | 
| 13 | 10, 12 | mpbir 231 | . . . 4 ⊢ ∅ Fn (∅ × ∅) | 
| 14 | 13 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅)) | 
| 15 | 4, 5, 6, 7, 14 | issubc2 17881 | . 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 713 | 1 ⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∅c0 4333 〈cop 4632 class class class wbr 5143 × cxp 5683 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 compcco 17309 Catccat 17707 Idccid 17708 Homf chomf 17709 ⊆cat cssc 17851 Subcatcsubc 17853 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-pm 8869 df-ixp 8938 df-homf 17713 df-ssc 17854 df-subc 17856 | 
| This theorem is referenced by: (None) | 
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