Theorem homf0 49049

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 2731	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
2		eqid 2731	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2731	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17596	. . 3 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦))
5		0mpo0 7429	. . . 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 2778	. 2 ⊢ ((Base‘𝐶) = ∅ → (Homf ‘𝐶) = ∅)
8	1, 2	homffn 17599	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
9		f0bi 6706	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ ↔ (Homf ‘𝐶) = ∅)
10		ffn 6651	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ → (Homf ‘𝐶) Fn ∅)
11	9, 10	sylbir 235	. . . 4 ⊢ ((Homf ‘𝐶) = ∅ → (Homf ‘𝐶) Fn ∅)
12		fndmu 6588	. . . 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 6107	. . . 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 4280   × cxp 5612   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Basecbs 17120  Hom chom 17172  Hom_f chomf 17572

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-homf 17576

This theorem is referenced by:  initopropdlem  49280  termopropdlem  49281