Theorem homf0 49787

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 2763	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
2		eqid 2763	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2763	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17741	. . 3 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦))
5		0mpo0 7493	. . . 4 ⊢ (((Base‘𝐶) = ∅ ∨ (Base‘𝐶) = ∅) → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦)) = ∅)
6	5	orcs 888	. . 3 ⊢ ((Base‘𝐶) = ∅ → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦)) = ∅)
7	4, 6	eqtrid 2810	. 2 ⊢ ((Base‘𝐶) = ∅ → (Homf ‘𝐶) = ∅)
8	1, 2	homffn 17744	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
9		f0bi 6761	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ ↔ (Homf ‘𝐶) = ∅)
10		ffn 6705	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ → (Homf ‘𝐶) Fn ∅)
11	9, 10	sylbir 238	. . . 4 ⊢ ((Homf ‘𝐶) = ∅ → (Homf ‘𝐶) Fn ∅)
12		fndmu 6642	. . . 4 ⊢ (((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ (Homf ‘𝐶) Fn ∅) → ((Base‘𝐶) × (Base‘𝐶)) = ∅)
13	8, 11, 12	sylancr 598	. . 3 ⊢ ((Homf ‘𝐶) = ∅ → ((Base‘𝐶) × (Base‘𝐶)) = ∅)
14		xpeq0 6157	. . . 4 ⊢ (((Base‘𝐶) × (Base‘𝐶)) = ∅ ↔ ((Base‘𝐶) = ∅ ∨ (Base‘𝐶) = ∅))
15		pm4.25 918	. . . 4 ⊢ ((Base‘𝐶) = ∅ ↔ ((Base‘𝐶) = ∅ ∨ (Base‘𝐶) = ∅))
16	14, 15	bitr4i 281	. . 3 ⊢ (((Base‘𝐶) × (Base‘𝐶)) = ∅ ↔ (Base‘𝐶) = ∅)
17	13, 16	sylib 221	. 2 ⊢ ((Homf ‘𝐶) = ∅ → (Base‘𝐶) = ∅)
18	7, 17	impbii 212	1 ⊢ ((Base‘𝐶) = ∅ ↔ (Homf ‘𝐶) = ∅)

Syntax hints:   ↔ wb 209   ∨ wo 860   = wceq 1570  ∅c0 4286   × cxp 5659   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Basecbs 17264  Hom chom 17316  Hom_f chomf 17717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-homf 17721

This theorem is referenced by:  initopropdlem  50018  termopropdlem  50019