Theorem 0funcALT 49557

Description: Alternate proof of 0func 49556. (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 17829	. 2 ⊢ Rel (∅ Func 𝐶)
2		0ex 5243	. . 3 ⊢ ∅ ∈ V
3	2, 2	relsnop 5761	. 2 ⊢ Rel {⟨∅, ∅⟩}
4		base0 17184	. . . . 5 ⊢ ∅ = (Base‘∅)
5		eqid 2737	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2737	. . . . 5 ⊢ (Hom ‘∅) = (Hom ‘∅)
7		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2737	. . . . 5 ⊢ (Id‘∅) = (Id‘∅)
9		eqid 2737	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		eqid 2737	. . . . 5 ⊢ (comp‘∅) = (comp‘∅)
11		eqid 2737	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
12		0cat 17655	. . . . . 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 17831	. . . 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 6724	. . . 4 ⊢ (𝑓:∅⟶(Base‘𝐶) ↔ 𝑓 = ∅)
17		ral0 4439	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑔𝑦):(𝑥(Hom ‘∅)𝑦)⟶((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦))
18	4	funcf2lem2 49551	. . . . . 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 711	. . . . 5 ⊢ (𝑔 ∈ X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st ‘𝑧))(Hom ‘𝐶)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ 𝑔 Fn (∅ × ∅))
20		0xp 5730	. . . . . 6 ⊢ (∅ × ∅) = ∅
21	20	fneq2i 6597	. . . . 5 ⊢ (𝑔 Fn (∅ × ∅) ↔ 𝑔 Fn ∅)
22		fn0 6630	. . . . 5 ⊢ (𝑔 Fn ∅ ↔ 𝑔 = ∅)
23	19, 21, 22	3bitri 297	. . . 4 ⊢ (𝑔 ∈ X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st ‘𝑧))(Hom ‘𝐶)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ 𝑔 = ∅)
24		ral0 4439	. . . 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 49555	. . 3 ⊢ (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
26		brsnop 5477	. . . 4 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (𝑓{⟨∅, ∅⟩}𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
27	2, 2, 26	mp2an 693	. . 3 ⊢ (𝑓{⟨∅, ∅⟩}𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅))
28	25, 27	bitr4di 289	. 2 ⊢ (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ 𝑓{⟨∅, ∅⟩}𝑔))
29	1, 3, 28	eqbrrdiv 5750	1 ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ∅c0 4274  {csn 4568  ⟨cop 4574   class class class wbr 5086   × cxp 5629   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-cnex 11094  ax-1cn 11096  ax-addcl 11098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-pss 3910  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-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  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-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-map 8775  df-ixp 8846  df-nn 12175  df-slot 17152  df-ndx 17164  df-base 17180  df-cat 17634  df-func 17825

This theorem is referenced by: (None)