Theorem 0ssc 17744

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 4351	. . 3 ⊢ ∅ ⊆ (Base‘𝐶)
2	1	a1i 11	. 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶))
3		ral0 4464	. . 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 6705	. . . . . 6 ⊢ ∅:∅⟶∅
6		ffn 6652	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
8		xp0 6107	. . . . . 6 ⊢ (∅ × ∅) = ∅
9	8	fneq2i 6580	. . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
10	7, 9	mpbir 231	. . . 4 ⊢ ∅ Fn (∅ × ∅)
11	10	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
12		eqid 2729	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
13		eqid 2729	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	12, 13	homffn 17599	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
15	14	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
16		fvexd 6837	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
17	11, 15, 16	isssc 17727	. 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 2109  ∀wral 3044  Vcvv 3436   ⊆ wss 3903  ∅c0 4284   class class class wbr 5092   × cxp 5617   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  Catccat 17570  Hom_f chomf 17572   ⊆_cat cssc 17714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-ixp 8825  df-homf 17576  df-ssc 17717

This theorem is referenced by:  0subcat  17745