Theorem 0ssc 17759

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 4350	. . 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 6713	. . . . . 6 ⊢ ∅:∅⟶∅
6		ffn 6660	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
8		xp0 5722	. . . . . 6 ⊢ (∅ × ∅) = ∅
9	8	fneq2i 6588	. . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
10	7, 9	mpbir 231	. . . 4 ⊢ ∅ Fn (∅ × ∅)
11	10	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
12		eqid 2734	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
13		eqid 2734	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	12, 13	homffn 17614	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
15	14	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
16		fvexd 6847	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
17	11, 15, 16	isssc 17742	. 2 ⊢ (𝐶 ∈ Cat → (∅ ⊆cat (Homf ‘𝐶) ↔ (∅ ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥∅𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))))
18	2, 4, 17	mpbir2and 713	1 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))

Syntax hints:   → wi 4   ∈ wcel 2113  ∀wral 3049  Vcvv 3438   ⊆ wss 3899  ∅c0 4283   class class class wbr 5096   × cxp 5620   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  Catccat 17585  Hom_f chomf 17587   ⊆_cat cssc 17729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-ixp 8834  df-homf 17591  df-ssc 17732

This theorem is referenced by:  0subcat  17760