Theorem 0funcALT 50118

Description: Alternate proof of 0func 50117. (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 17999	. 2 ⊢ Rel (∅ Func 𝐶)
2		0ex 5260	. . 3 ⊢ ∅ ∈ V
3	2, 2	relsnop 5779	. 2 ⊢ Rel {⟨∅, ∅⟩}
4		base0 17354	. . . . 5 ⊢ ∅ = (Base‘∅)
5		eqid 2760	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2760	. . . . 5 ⊢ (Hom ‘∅) = (Hom ‘∅)
7		eqid 2760	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2760	. . . . 5 ⊢ (Id‘∅) = (Id‘∅)
9		eqid 2760	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		eqid 2760	. . . . 5 ⊢ (comp‘∅) = (comp‘∅)
11		eqid 2760	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
12		0cat 17825	. . . . . 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 18001	. . . 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 6753	. . . 4 ⊢ (𝑓:∅⟶(Base‘𝐶) ↔ 𝑓 = ∅)
17		ral0 4453	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑔𝑦):(𝑥(Hom ‘∅)𝑦)⟶((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦))
18	4	funcf2lem2 50112	. . . . . 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 5746	. . . . . 6 ⊢ (∅ × ∅) = ∅
21	20	fneq2i 6625	. . . . 5 ⊢ (𝑔 Fn (∅ × ∅) ↔ 𝑔 Fn ∅)
22		fn0 6658	. . . . 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 4453	. . . 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 50116	. . 3 ⊢ (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
26		brsnop 5492	. . . 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 5766	1 ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  ∅c0 4278  {csn 4583  ⟨cop 4589   class class class wbr 5102   × cxp 5645   Fn wfn 6522  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408  1^st c1st 7982  2^nd c2nd 7983   ↑_m cmap 8825  Xcixp 8903  Basecbs 17349  Hom chom 17401  compcco 17402  Catccat 17800  Idccid 17801   Func cfunc 17991

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11228  ax-1cn 11230  ax-addcl 11232

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-map 8827  df-ixp 8904  df-nn 12306  df-slot 17322  df-ndx 17334  df-base 17350  df-cat 17804  df-func 17995

This theorem is used by: (None)