Theorem 0funcg2 50530

Description: The functor from the empty category. (Contributed by Zhi Wang, 17-Oct-2025.)

Hypotheses

Ref	Expression
0funcg.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
0funcg.b	⊢ (𝜑 → ∅ = (Base‘𝐶))
0funcg.d	⊢ (𝜑 → 𝐷 ∈ Cat)

Assertion

Ref	Expression
0funcg2	⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ (𝐹 = ∅ ∧ 𝐺 = ∅)))

Proof of Theorem 0funcg2

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

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2763	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2763	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2763	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2763	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
6		eqid 2763	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2763	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
8		eqid 2763	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		0funcg.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
10		0funcg.b	. . . 4 ⊢ (𝜑 → ∅ = (Base‘𝐶))
11		0catg 17912	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)
12	9, 10, 11	syl2anc 596	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		0funcg.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
14	1, 2, 3, 4, 5, 6, 7, 8, 12, 13	isfunc 18089	. 2 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ (𝐹:(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝐺 ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ∧ ∀𝑥 ∈ (Base‘𝐶)(((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑚 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
15	10	feq2d 6723	. . 3 ⊢ (𝜑 → (𝐹:∅⟶(Base‘𝐷) ↔ 𝐹:(Base‘𝐶)⟶(Base‘𝐷)))
16		f0bi 6795	. . 3 ⊢ (𝐹:∅⟶(Base‘𝐷) ↔ 𝐹 = ∅)
17	15, 16	bitr3di 289	. 2 ⊢ (𝜑 → (𝐹:(Base‘𝐶)⟶(Base‘𝐷) ↔ 𝐹 = ∅))
18	10	eqcomd 2769	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = ∅)
19		rzal 4453	. . . . 5 ⊢ ((Base‘𝐶) = ∅ → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
20	18, 19	syl 18	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
21	1	funcf2lem2 50527	. . . . 5 ⊢ (𝐺 ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ↔ (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))))
22	21	a1i 11	. . . 4 ⊢ (𝜑 → (𝐺 ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ↔ (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))))
23	20, 22	mpbiran2d 721	. . 3 ⊢ (𝜑 → (𝐺 ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ↔ 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶))))
24	10	sqxpeqd 5690	. . . . . 6 ⊢ (𝜑 → (∅ × ∅) = ((Base‘𝐶) × (Base‘𝐶)))
25		0xp 5758	. . . . . 6 ⊢ (∅ × ∅) = ∅
26	24, 25	eqtr3di 2813	. . . . 5 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ∅)
27	26	fneq2d 6663	. . . 4 ⊢ (𝜑 → (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ 𝐺 Fn ∅))
28		fn0 6700	. . . 4 ⊢ (𝐺 Fn ∅ ↔ 𝐺 = ∅)
29	27, 28	bitrdi 290	. . 3 ⊢ (𝜑 → (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ 𝐺 = ∅))
30	23, 29	bitrd 282	. 2 ⊢ (𝜑 → (𝐺 ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ↔ 𝐺 = ∅))
31		rzal 4453	. . 3 ⊢ ((Base‘𝐶) = ∅ → ∀𝑥 ∈ (Base‘𝐶)(((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑚 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
32	18, 31	syl 18	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)(((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑚 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
33	14, 17, 30, 32	0funcglem 50529	1 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ (𝐹 = ∅ ∧ 𝐺 = ∅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∅c0 4281  ⟨cop 4840   class class class wbr 5108   × cxp 5656   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034   ↑_m cmap 8885  Xcixp 8963  Basecbs 17431  Hom chom 17483  compcco 17484  Catccat 17888  Idccid 17889   Func cfunc 18079

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887  df-ixp 8964  df-cat 17892  df-func 18083

This theorem is used by:  0funcg  50531  termolmd  51184