Theorem 0funcALT 49657

Description: Alternate proof of 0func 49656. (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 17871	. 2 ⊢ Rel (∅ Func 𝐶)
2		0ex 5251	. . 3 ⊢ ∅ ∈ V
3	2, 2	relsnop 5771	. 2 ⊢ Rel {⟨∅, ∅⟩}
4		base0 17226	. . . . 5 ⊢ ∅ = (Base‘∅)
5		eqid 2756	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2756	. . . . 5 ⊢ (Hom ‘∅) = (Hom ‘∅)
7		eqid 2756	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2756	. . . . 5 ⊢ (Id‘∅) = (Id‘∅)
9		eqid 2756	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		eqid 2756	. . . . 5 ⊢ (comp‘∅) = (comp‘∅)
11		eqid 2756	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
12		0cat 17697	. . . . . 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 17873	. . . 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 6736	. . . 4 ⊢ (𝑓:∅⟶(Base‘𝐶) ↔ 𝑓 = ∅)
17		ral0 4446	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑔𝑦):(𝑥(Hom ‘∅)𝑦)⟶((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦))
18	4	funcf2lem2 49651	. . . . . 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 718	. . . . 5 ⊢ (𝑔 ∈ X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st ‘𝑧))(Hom ‘𝐶)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ 𝑔 Fn (∅ × ∅))
20		0xp 5739	. . . . . 6 ⊢ (∅ × ∅) = ∅
21	20	fneq2i 6608	. . . . 5 ⊢ (𝑔 Fn (∅ × ∅) ↔ 𝑔 Fn ∅)
22		fn0 6641	. . . . 5 ⊢ (𝑔 Fn ∅ ↔ 𝑔 = ∅)
23	19, 21, 22	3bitri 299	. . . 4 ⊢ (𝑔 ∈ X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st ‘𝑧))(Hom ‘𝐶)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ 𝑔 = ∅)
24		ral0 4446	. . . 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 49655	. . 3 ⊢ (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
26		brsnop 5486	. . . 4 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (𝑓{⟨∅, ∅⟩}𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
27	2, 2, 26	mp2an 700	. . 3 ⊢ (𝑓{⟨∅, ∅⟩}𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅))
28	25, 27	bitr4di 291	. 2 ⊢ (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ 𝑓{⟨∅, ∅⟩}𝑔))
29	1, 3, 28	eqbrrdiv 5759	1 ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∀wral 3070  Vcvv 3448  ∅c0 4280  {csn 4576  ⟨cop 4582   class class class wbr 5094   × cxp 5638   Fn wfn 6505  ⟶wf 6506  ‘cfv 6510  (class class class)co 7385  1^st c1st 7957  2^nd c2nd 7958   ↑_m cmap 8796  Xcixp 8868  Basecbs 17221  Hom chom 17273  compcco 17274  Catccat 17672  Idccid 17673   Func cfunc 17863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-cnex 11119  ax-1cn 11121  ax-addcl 11123

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-map 8798  df-ixp 8869  df-nn 12201  df-slot 17194  df-ndx 17206  df-base 17222  df-cat 17676  df-func 17867

This theorem is referenced by: (None)