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Mirrors > Home > MPE Home > Th. List > 0ssc | Structured version Visualization version GIF version |
Description: For any category 𝐶, the empty set is a subcategory subset of 𝐶. (Contributed by AV, 23-Apr-2020.) |
Ref | Expression |
---|---|
0ssc | ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4347 | . . 3 ⊢ ∅ ⊆ (Base‘𝐶) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶)) |
3 | ral0 4452 | . . 3 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥∅𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦) | |
4 | 3 | a1i 11 | . 2 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥∅𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦)) |
5 | f0 6553 | . . . . . 6 ⊢ ∅:∅⟶∅ | |
6 | ffn 6507 | . . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ ∅ Fn ∅ |
8 | xp0 6008 | . . . . . 6 ⊢ (∅ × ∅) = ∅ | |
9 | 8 | fneq2i 6444 | . . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅) |
10 | 7, 9 | mpbir 232 | . . . 4 ⊢ ∅ Fn (∅ × ∅) |
11 | 10 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅)) |
12 | eqid 2818 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
13 | eqid 2818 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
14 | 12, 13 | homffn 16951 | . . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
15 | 14 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) |
16 | fvexd 6678 | . . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V) | |
17 | 11, 15, 16 | isssc 17078 | . 2 ⊢ (𝐶 ∈ Cat → (∅ ⊆cat (Homf ‘𝐶) ↔ (∅ ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥∅𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦)))) |
18 | 2, 4, 17 | mpbir2and 709 | 1 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 class class class wbr 5057 × cxp 5546 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Catccat 16923 Homf chomf 16925 ⊆cat cssc 17065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-ixp 8450 df-homf 16929 df-ssc 17068 |
This theorem is referenced by: 0subcat 17096 |
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