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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 4336 | . . 3 ⊢ ∅ ⊆ (Base‘𝐶) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶)) |
3 | ral0 4449 | . . 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 6653 | . . . . . 6 ⊢ ∅:∅⟶∅ | |
6 | ffn 6598 | . . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ ∅ Fn ∅ |
8 | xp0 6060 | . . . . . 6 ⊢ (∅ × ∅) = ∅ | |
9 | 8 | fneq2i 6529 | . . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅) |
10 | 7, 9 | mpbir 230 | . . . 4 ⊢ ∅ Fn (∅ × ∅) |
11 | 10 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅)) |
12 | eqid 2740 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
13 | eqid 2740 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
14 | 12, 13 | homffn 17400 | . . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
15 | 14 | a1i 11 | . . 3 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) |
16 | fvexd 6786 | . . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V) | |
17 | 11, 15, 16 | isssc 17530 | . 2 ⊢ (𝐶 ∈ Cat → (∅ ⊆cat (Homf ‘𝐶) ↔ (∅ ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥∅𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦)))) |
18 | 2, 4, 17 | mpbir2and 710 | 1 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∀wral 3066 Vcvv 3431 ⊆ wss 3892 ∅c0 4262 class class class wbr 5079 × cxp 5588 Fn wfn 6427 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 Catccat 17371 Homf chomf 17373 ⊆cat cssc 17517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-1st 7824 df-2nd 7825 df-ixp 8669 df-homf 17377 df-ssc 17520 |
This theorem is referenced by: 0subcat 17551 |
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