Theorem 0funcALT 50408

Description: Alternate proof of 0func 50407. (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 18066	. 2 ⊢ Rel (∅ Func 𝐶)
2		0ex 5264	. . 3 ⊢ ∅ ∈ V
3	2, 2	relsnop 5788	. 2 ⊢ Rel {⟨∅, ∅⟩}
4		base0 17421	. . . . 5 ⊢ ∅ = (Base‘∅)
5		eqid 2763	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2763	. . . . 5 ⊢ (Hom ‘∅) = (Hom ‘∅)
7		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2763	. . . . 5 ⊢ (Id‘∅) = (Id‘∅)
9		eqid 2763	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		eqid 2763	. . . . 5 ⊢ (comp‘∅) = (comp‘∅)
11		eqid 2763	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
12		0cat 17892	. . . . . 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 18068	. . . 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 6787	. . . 4 ⊢ (𝑓:∅⟶(Base‘𝐶) ↔ 𝑓 = ∅)
17		ral0 4457	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑔𝑦):(𝑥(Hom ‘∅)𝑦)⟶((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦))
18	4	funcf2lem2 50401	. . . . . 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 5755	. . . . . 6 ⊢ (∅ × ∅) = ∅
21	20	fneq2i 6659	. . . . 5 ⊢ (𝑔 Fn (∅ × ∅) ↔ 𝑔 Fn ∅)
22		fn0 6692	. . . . 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 50406	. . 3 ⊢ (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
26		brsnop 5500	. . . 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 5775	1 ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ∅c0 4281  {csn 4588  ⟨cop 4838   class class class wbr 5106   × cxp 5653   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024   ↑_m cmap 8875  Xcixp 8953  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Idccid 17868   Func cfunc 18058

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-1cn 11285  ax-addcl 11287

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 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-pss 4017  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 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-map 8877  df-ixp 8954  df-nn 12363  df-slot 17389  df-ndx 17401  df-base 17417  df-cat 17871  df-func 18062

This theorem is used by: (None)