Theorem homf0 49196

Description: The base is empty iff the functionalized Hom-set operation is empty. (Contributed by Zhi Wang, 23-Oct-2025.)

Assertion

Ref	Expression
homf0	⊢ ((Base‘𝐶) = ∅ ↔ (Homf ‘𝐶) = ∅)

Proof of Theorem homf0

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

Step	Hyp	Ref	Expression
1		eqid 2734	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
2		eqid 2734	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2734	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17611	. . 3 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦))
5		0mpo0 7439	. . . 4 ⊢ (((Base‘𝐶) = ∅ ∨ (Base‘𝐶) = ∅) → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦)) = ∅)
6	5	orcs 875	. . 3 ⊢ ((Base‘𝐶) = ∅ → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦)) = ∅)
7	4, 6	eqtrid 2781	. 2 ⊢ ((Base‘𝐶) = ∅ → (Homf ‘𝐶) = ∅)
8	1, 2	homffn 17614	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
9		f0bi 6715	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ ↔ (Homf ‘𝐶) = ∅)
10		ffn 6660	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ → (Homf ‘𝐶) Fn ∅)
11	9, 10	sylbir 235	. . . 4 ⊢ ((Homf ‘𝐶) = ∅ → (Homf ‘𝐶) Fn ∅)
12		fndmu 6597	. . . 4 ⊢ (((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ (Homf ‘𝐶) Fn ∅) → ((Base‘𝐶) × (Base‘𝐶)) = ∅)
13	8, 11, 12	sylancr 587	. . 3 ⊢ ((Homf ‘𝐶) = ∅ → ((Base‘𝐶) × (Base‘𝐶)) = ∅)
14		xpeq0 6116	. . . 4 ⊢ (((Base‘𝐶) × (Base‘𝐶)) = ∅ ↔ ((Base‘𝐶) = ∅ ∨ (Base‘𝐶) = ∅))
15		pm4.25 905	. . . 4 ⊢ ((Base‘𝐶) = ∅ ↔ ((Base‘𝐶) = ∅ ∨ (Base‘𝐶) = ∅))
16	14, 15	bitr4i 278	. . 3 ⊢ (((Base‘𝐶) × (Base‘𝐶)) = ∅ ↔ (Base‘𝐶) = ∅)
17	13, 16	sylib 218	. 2 ⊢ ((Homf ‘𝐶) = ∅ → (Base‘𝐶) = ∅)
18	7, 17	impbii 209	1 ⊢ ((Base‘𝐶) = ∅ ↔ (Homf ‘𝐶) = ∅)

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1541  ∅c0 4283   × cxp 5620   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  Basecbs 17134  Hom chom 17186  Hom_f chomf 17587

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-homf 17591

This theorem is referenced by:  initopropdlem  49427  termopropdlem  49428