Theorem 0funcALT 50016

Description: Alternate proof of 0func 50015. (Contributed by Zhi Wang, 7-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
0func.c	⊢ (𝜑 → 𝐶 ∈ Cat)

Assertion

Ref	Expression
0funcALT	⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩})

Proof of Theorem 0funcALT

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

Step	Hyp	Ref	Expression
1		relfunc 17955	. 2 ⊢ Rel (∅ Func 𝐶)
2		0ex 5268	. . 3 ⊢ ∅ ∈ V
3	2, 2	relsnop 5790	. 2 ⊢ Rel {⟨∅, ∅⟩}
4		base0 17310	. . . . 5 ⊢ ∅ = (Base‘∅)
5		eqid 2762	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2762	. . . . 5 ⊢ (Hom ‘∅) = (Hom ‘∅)
7		eqid 2762	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2762	. . . . 5 ⊢ (Id‘∅) = (Id‘∅)
9		eqid 2762	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		eqid 2762	. . . . 5 ⊢ (comp‘∅) = (comp‘∅)
11		eqid 2762	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
12		0cat 17781	. . . . . 6 ⊢ ∅ ∈ Cat
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ Cat)
14		0func.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
15	4, 5, 6, 7, 8, 9, 10, 11, 13, 14	isfunc 17957	. . . 4 ⊢ (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ (𝑓:∅⟶(Base‘𝐶) ∧ 𝑔 ∈ X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st ‘𝑧))(Hom ‘𝐶)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ∧ ∀𝑥 ∈ ∅ (((𝑥𝑔𝑥)‘((Id‘∅)‘𝑥)) = ((Id‘𝐶)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑚 ∈ (𝑥(Hom ‘∅)𝑦)∀𝑛 ∈ (𝑦(Hom ‘∅)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘∅)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝐶)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))))
16		f0bi 6762	. . . 4 ⊢ (𝑓:∅⟶(Base‘𝐶) ↔ 𝑓 = ∅)
17		ral0 4457	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑔𝑦):(𝑥(Hom ‘∅)𝑦)⟶((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦))
18	4	funcf2lem2 50010	. . . . . 6 ⊢ (𝑔 ∈ X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st ‘𝑧))(Hom ‘𝐶)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ (𝑔 Fn (∅ × ∅) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑔𝑦):(𝑥(Hom ‘∅)𝑦)⟶((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦))))
19	17, 18	mpbiran2 723	. . . . 5 ⊢ (𝑔 ∈ X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st ‘𝑧))(Hom ‘𝐶)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ 𝑔 Fn (∅ × ∅))
20		0xp 5758	. . . . . 6 ⊢ (∅ × ∅) = ∅
21	20	fneq2i 6634	. . . . 5 ⊢ (𝑔 Fn (∅ × ∅) ↔ 𝑔 Fn ∅)
22		fn0 6667	. . . . 5 ⊢ (𝑔 Fn ∅ ↔ 𝑔 = ∅)
23	19, 21, 22	3bitri 300	. . . 4 ⊢ (𝑔 ∈ X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st ‘𝑧))(Hom ‘𝐶)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ 𝑔 = ∅)
24		ral0 4457	. . . 4 ⊢ ∀𝑥 ∈ ∅ (((𝑥𝑔𝑥)‘((Id‘∅)‘𝑥)) = ((Id‘𝐶)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑚 ∈ (𝑥(Hom ‘∅)𝑦)∀𝑛 ∈ (𝑦(Hom ‘∅)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘∅)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝐶)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))
25	15, 16, 23, 24	0funclem 50014	. . 3 ⊢ (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
26		brsnop 5504	. . . 4 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (𝑓{⟨∅, ∅⟩}𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
27	2, 2, 26	mp2an 705	. . 3 ⊢ (𝑓{⟨∅, ∅⟩}𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅))
28	25, 27	bitr4di 292	. 2 ⊢ (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ 𝑓{⟨∅, ∅⟩}𝑔))
29	1, 3, 28	eqbrrdiv 5778	1 ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453  ∅c0 4282  {csn 4587  ⟨cop 4593   class class class wbr 5107   × cxp 5657   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  1^st c1st 7987  2^nd c2nd 7988   ↑_m cmap 8829  Xcixp 8907  Basecbs 17305  Hom chom 17357  compcco 17358  Catccat 17756  Idccid 17757   Func cfunc 17947

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-1cn 11185  ax-addcl 11187

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  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 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-map 8831  df-ixp 8908  df-nn 12261  df-slot 17278  df-ndx 17290  df-base 17306  df-cat 17760  df-func 17951

This theorem is used by: (None)