Theorem homf0 49499

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 2737	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
2		eqid 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2737	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17650	. . 3 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦))
5		0mpo0 7444	. . . 4 ⊢ (((Base‘𝐶) = ∅ ∨ (Base‘𝐶) = ∅) → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦)) = ∅)
6	5	orcs 876	. . 3 ⊢ ((Base‘𝐶) = ∅ → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(Hom ‘𝐶)𝑦)) = ∅)
7	4, 6	eqtrid 2784	. 2 ⊢ ((Base‘𝐶) = ∅ → (Homf ‘𝐶) = ∅)
8	1, 2	homffn 17653	. . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
9		f0bi 6718	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ ↔ (Homf ‘𝐶) = ∅)
10		ffn 6663	. . . . 5 ⊢ ((Homf ‘𝐶):∅⟶∅ → (Homf ‘𝐶) Fn ∅)
11	9, 10	sylbir 235	. . . 4 ⊢ ((Homf ‘𝐶) = ∅ → (Homf ‘𝐶) Fn ∅)
12		fndmu 6600	. . . 4 ⊢ (((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ (Homf ‘𝐶) Fn ∅) → ((Base‘𝐶) × (Base‘𝐶)) = ∅)
13	8, 11, 12	sylancr 588	. . 3 ⊢ ((Homf ‘𝐶) = ∅ → ((Base‘𝐶) × (Base‘𝐶)) = ∅)
14		xpeq0 6119	. . . 4 ⊢ (((Base‘𝐶) × (Base‘𝐶)) = ∅ ↔ ((Base‘𝐶) = ∅ ∨ (Base‘𝐶) = ∅))
15		pm4.25 906	. . . 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 848   = wceq 1542  ∅c0 4274   × cxp 5623   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363  Basecbs 17173  Hom chom 17225  Hom_f chomf 17626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-homf 17630

This theorem is referenced by:  initopropdlem  49730  termopropdlem  49731