Theorem 0ssc 17795

Description: For any category 𝐶, the empty set is a subcategory subset of 𝐶. (Contributed by AV, 23-Apr-2020.)

Assertion

Ref	Expression
0ssc	⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))

Proof of Theorem 0ssc

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ss 4328	. . 3 ⊢ ∅ ⊆ (Base‘𝐶)
2	1	a1i 11	. 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶))
3		ral0 4426	. . 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 6708	. . . . . 6 ⊢ ∅:∅⟶∅
6		ffn 6655	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
8		xp0 5718	. . . . . 6 ⊢ (∅ × ∅) = ∅
9	8	fneq2i 6583	. . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
10	7, 9	mpbir 232	. . . 4 ⊢ ∅ Fn (∅ × ∅)
11	10	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
12		eqid 2739	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
13		eqid 2739	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	12, 13	homffn 17650	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
15	14	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
16		fvexd 6842	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
17	11, 15, 16	isssc 17778	. 2 ⊢ (𝐶 ∈ Cat → (∅ ⊆cat (Homf ‘𝐶) ↔ (∅ ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥∅𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))))
18	2, 4, 17	mpbir2and 719	1 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))

Syntax hints:   → wi 4   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ⊆ wss 3883  ∅c0 4261   class class class wbr 5072   × cxp 5616   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  Catccat 17621  Hom_f chomf 17623   ⊆_cat cssc 17765

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-ixp 8836  df-homf 17627  df-ssc 17768

This theorem is referenced by:  0subcat  17796