Theorem 0ssc 16320

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 3924	. . 3 ⊢ ∅ ⊆ (Base‘𝐶)
2	1	a1i 11	. 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶))
3		ral0 4028	. . 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 5999	. . . . . 6 ⊢ ∅:∅⟶∅
6		ffn 5958	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
8		xp0 5471	. . . . . 6 ⊢ (∅ × ∅) = ∅
9	8	fneq2i 5900	. . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
10	7, 9	mpbir 220	. . . 4 ⊢ ∅ Fn (∅ × ∅)
11	10	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
12		eqid 2610	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
13		eqid 2610	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	12, 13	homffn 16176	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
15	14	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
16		fvex 6113	. . . 4 ⊢ (Base‘𝐶) ∈ V
17	16	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
18	11, 15, 17	isssc 16303	. 2 ⊢ (𝐶 ∈ Cat → (∅ ⊆cat (Homf ‘𝐶) ↔ (∅ ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥∅𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))))
19	2, 4, 18	mpbir2and 959	1 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))

Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   × cxp 5036   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Catccat 16148  Hom_f chomf 16150   ⊆_cat cssc 16290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-ixp 7795  df-homf 16154  df-ssc 16293

This theorem is referenced by:  0subcat  16321