Theorem homf0 49671

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 2769	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
2		eqid 2769	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2769	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17745	. . 3 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦))
5		0mpo0 7494	. . . 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 2816	. 2 ⊢ ((Base‘𝐶) = ∅ → (Homf ‘𝐶) = ∅)
8	1, 2	homffn 17748	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
9		f0bi 6762	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ ↔ (Homf ‘𝐶) = ∅)
10		ffn 6706	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ → (Homf ‘𝐶) Fn ∅)
11	9, 10	sylbir 238	. . . 4 ⊢ ((Homf ‘𝐶) = ∅ → (Homf ‘𝐶) Fn ∅)
12		fndmu 6643	. . . 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 6158	. . . 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 1567  ∅c0 4294   × cxp 5660   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17268  Hom chom 17320  Hom_f chomf 17721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-homf 17725

This theorem is referenced by:  initopropdlem  49902  termopropdlem  49903