Theorem 0funcg2 49559

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 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2736	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2736	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2736	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2736	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
6		eqid 2736	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2736	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
8		eqid 2736	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		0funcg.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
10		0funcg.b	. . . 4 ⊢ (𝜑 → ∅ = (Base‘𝐶))
11		0catg 17654	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)
12	9, 10, 11	syl2anc 585	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		0funcg.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
14	1, 2, 3, 4, 5, 6, 7, 8, 12, 13	isfunc 17831	. 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 6652	. . 3 ⊢ (𝜑 → (𝐹:∅⟶(Base‘𝐷) ↔ 𝐹:(Base‘𝐶)⟶(Base‘𝐷)))
16		f0bi 6723	. . 3 ⊢ (𝐹:∅⟶(Base‘𝐷) ↔ 𝐹 = ∅)
17	15, 16	bitr3di 286	. 2 ⊢ (𝜑 → (𝐹:(Base‘𝐶)⟶(Base‘𝐷) ↔ 𝐹 = ∅))
18	10	eqcomd 2742	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = ∅)
19		rzal 4434	. . . . 5 ⊢ ((Base‘𝐶) = ∅ → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
20	18, 19	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
21	1	funcf2lem2 49557	. . . . 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 709	. . 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 5663	. . . . . 6 ⊢ (𝜑 → (∅ × ∅) = ((Base‘𝐶) × (Base‘𝐶)))
25		0xp 5730	. . . . . 6 ⊢ (∅ × ∅) = ∅
26	24, 25	eqtr3di 2786	. . . . 5 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ∅)
27	26	fneq2d 6592	. . . 4 ⊢ (𝜑 → (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ 𝐺 Fn ∅))
28		fn0 6629	. . . 4 ⊢ (𝐺 Fn ∅ ↔ 𝐺 = ∅)
29	27, 28	bitrdi 287	. . 3 ⊢ (𝜑 → (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ 𝐺 = ∅))
30	23, 29	bitrd 279	. 2 ⊢ (𝜑 → (𝐺 ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ↔ 𝐺 = ∅))
31		rzal 4434	. . 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 17	. 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 49558	1 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ (𝐹 = ∅ ∧ 𝐺 = ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∅c0 4273  ⟨cop 4573   class class class wbr 5085   × cxp 5629   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941   ↑_m cmap 8773  Xcixp 8845  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17630  Idccid 17631   Func cfunc 17821

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-cat 17634  df-func 17825

This theorem is referenced by:  0funcg  49560  termolmd  50145