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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 4333 | . . 3 ⊢ ∅ ⊆ (Base‘𝐶) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶)) |
3 | ral0 4446 | . . 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 6673 | . . . . . 6 ⊢ ∅:∅⟶∅ | |
6 | ffn 6618 | . . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ ∅ Fn ∅ |
8 | xp0 6065 | . . . . . 6 ⊢ (∅ × ∅) = ∅ | |
9 | 8 | fneq2i 6550 | . . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅) |
10 | 7, 9 | mpbir 230 | . . . 4 ⊢ ∅ Fn (∅ × ∅) |
11 | 10 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅)) |
12 | eqid 2733 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
13 | eqid 2733 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
14 | 12, 13 | homffn 17430 | . . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
15 | 14 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) |
16 | fvexd 6807 | . . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V) | |
17 | 11, 15, 16 | isssc 17560 | . 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 2101 ∀wral 3059 Vcvv 3434 ⊆ wss 3889 ∅c0 4259 class class class wbr 5077 × cxp 5589 Fn wfn 6442 ⟶wf 6443 ‘cfv 6447 (class class class)co 7295 Basecbs 16940 Catccat 17401 Homf chomf 17403 ⊆cat cssc 17547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-1st 7851 df-2nd 7852 df-ixp 8706 df-homf 17407 df-ssc 17550 |
This theorem is referenced by: 0subcat 17581 |
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