Theorem 0ssc 17806

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 4366	. . 3 ⊢ ∅ ⊆ (Base‘𝐶)
2	1	a1i 11	. 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶))
3		ral0 4479	. . 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 6744	. . . . . 6 ⊢ ∅:∅⟶∅
6		ffn 6691	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
8		xp0 6134	. . . . . 6 ⊢ (∅ × ∅) = ∅
9	8	fneq2i 6619	. . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
10	7, 9	mpbir 231	. . . 4 ⊢ ∅ Fn (∅ × ∅)
11	10	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
12		eqid 2730	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
13		eqid 2730	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	12, 13	homffn 17661	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
15	14	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
16		fvexd 6876	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
17	11, 15, 16	isssc 17789	. 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 3045  Vcvv 3450   ⊆ wss 3917  ∅c0 4299   class class class wbr 5110   × cxp 5639   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Catccat 17632  Hom_f chomf 17634   ⊆_cat cssc 17776

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-ixp 8874  df-homf 17638  df-ssc 17779

This theorem is referenced by:  0subcat  17807