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Theorem 0funcALT 49750
Description: Alternate proof of 0func 49749. (Contributed by Zhi Wang, 7-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
0func.c (𝜑𝐶 ∈ 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.
StepHypRef Expression
1 relfunc 17918 . 2 Rel (∅ Func 𝐶)
2 0ex 5272 . . 3 ∅ ∈ V
32, 2relsnop 5793 . 2 Rel {⟨∅, ∅⟩}
4 base0 17273 . . . . 5 ∅ = (Base‘∅)
5 eqid 2769 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 eqid 2769 . . . . 5 (Hom ‘∅) = (Hom ‘∅)
7 eqid 2769 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
8 eqid 2769 . . . . 5 (Id‘∅) = (Id‘∅)
9 eqid 2769 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
10 eqid 2769 . . . . 5 (comp‘∅) = (comp‘∅)
11 eqid 2769 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
12 0cat 17744 . . . . . 6 ∅ ∈ Cat
1312a1i 11 . . . . 5 (𝜑 → ∅ ∈ Cat)
14 0func.c . . . . 5 (𝜑𝐶 ∈ Cat)
154, 5, 6, 7, 8, 9, 10, 11, 13, 14isfunc 17920 . . . 4 (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ (𝑓:∅⟶(Base‘𝐶) ∧ 𝑔X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st𝑧))(Hom ‘𝐶)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ∧ ∀𝑥 ∈ ∅ (((𝑥𝑔𝑥)‘((Id‘∅)‘𝑥)) = ((Id‘𝐶)‘(𝑓𝑥)) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑚 ∈ (𝑥(Hom ‘∅)𝑦)∀𝑛 ∈ (𝑦(Hom ‘∅)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘∅)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝐶)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))))
16 f0bi 6762 . . . 4 (𝑓:∅⟶(Base‘𝐶) ↔ 𝑓 = ∅)
17 ral0 4464 . . . . . 6 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑔𝑦):(𝑥(Hom ‘∅)𝑦)⟶((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦))
184funcf2lem2 49744 . . . . . 6 (𝑔X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st𝑧))(Hom ‘𝐶)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ (𝑔 Fn (∅ × ∅) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑔𝑦):(𝑥(Hom ‘∅)𝑦)⟶((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦))))
1917, 18mpbiran2 722 . . . . 5 (𝑔X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st𝑧))(Hom ‘𝐶)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ 𝑔 Fn (∅ × ∅))
20 0xp 5761 . . . . . 6 (∅ × ∅) = ∅
2120fneq2i 6634 . . . . 5 (𝑔 Fn (∅ × ∅) ↔ 𝑔 Fn ∅)
22 fn0 6667 . . . . 5 (𝑔 Fn ∅ ↔ 𝑔 = ∅)
2319, 21, 223bitri 300 . . . 4 (𝑔X𝑧 ∈ (∅ × ∅)(((𝑓‘(1st𝑧))(Hom ‘𝐶)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘∅)‘𝑧)) ↔ 𝑔 = ∅)
24 ral0 4464 . . . 4 𝑥 ∈ ∅ (((𝑥𝑔𝑥)‘((Id‘∅)‘𝑥)) = ((Id‘𝐶)‘(𝑓𝑥)) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑚 ∈ (𝑥(Hom ‘∅)𝑦)∀𝑛 ∈ (𝑦(Hom ‘∅)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘∅)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝐶)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))
2515, 16, 23, 240funclem 49748 . . 3 (𝜑 → (𝑓(∅ Func 𝐶)𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
26 brsnop 5507 . . . 4 ((∅ ∈ V ∧ ∅ ∈ V) → (𝑓{⟨∅, ∅⟩}𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅)))
272, 2, 26mp2an 704 . . 3 (𝑓{⟨∅, ∅⟩}𝑔 ↔ (𝑓 = ∅ ∧ 𝑔 = ∅))
2825, 27bitr4di 292 . 2 (𝜑 → (𝑓(∅ Func 𝐶)𝑔𝑓{⟨∅, ∅⟩}𝑔))
291, 3, 28eqbrrdiv 5781 1 (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 400   = wceq 1567  wcel 2149  wral 3085  Vcvv 3463  c0 4294  {csn 4594  cop 4600   class class class wbr 5113   × cxp 5660   Fn wfn 6532  wf 6533  cfv 6537  (class class class)co 7411  1st c1st 7983  2nd c2nd 7984  m cmap 8823  Xcixp 8894  Basecbs 17268  Hom chom 17320  compcco 17321  Catccat 17719  Idccid 17720   Func cfunc 17910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-1cn 11157  ax-addcl 11159
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-map 8825  df-ixp 8895  df-nn 12233  df-slot 17241  df-ndx 17253  df-base 17269  df-cat 17723  df-func 17914
This theorem is referenced by: (None)
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