Theorem fuconuniqvals 50989

Description: At the same natural-transformation/identity input, the canonical and alternative morphism maps give respectively ∅ and 1_o at the unique source object ∅. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

Hypotheses

Ref	Expression
fuconuniq2.c	⊢ 𝐶 = (SetCat‘1o)
fuconuniq2.e	⊢ 𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}
fuconuniq2.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fuconuniq2.l	⊢ 𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)
fuconuniq.a	⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
fuconuniq.m	⊢ 𝑀 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))
fuconuniq.s	⊢ 𝑆 = ⟨( I ↾ 2o), 𝑀⟩
fuconuniq.r	⊢ 𝑅 = (𝐸 FuncCat 𝐸)
fuconuniq.o	⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))
fuconuniq.k	⊢ 𝐾 = ((1st ‘ ⚬ )‘𝑆)
fuconuniqvals.d	⊢ 𝐷 = (𝐸Δfunc𝐶)
fuconuniqvals.f	⊢ 𝐹 = ((1st ‘𝐷)‘∅)
fuconuniqvals.g	⊢ 𝐺 = ((1st ‘𝐷)‘1o)
fuconuniqvals.n	⊢ 𝑁 = (1o × {∅})

Assertion

Ref	Expression
fuconuniqvals	⊢ (((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = ∅ ∧ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = 1o)

Distinct variable groups:   𝑚,𝑛,𝑟,𝑠,𝑝,𝑞,𝑢,𝑣,𝑎,𝑏   𝐴,𝑚,𝑛,𝑟,𝑠,𝑝,𝑞,𝑢,𝑣,𝑎

Allowed substitution hints:   𝐴(𝑏)   𝐶(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐷(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑄(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑅(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑆(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐸(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐹(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐺(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐾(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐿(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑀(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑁(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   ⚬ (𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)

Proof of Theorem fuconuniqvals

Step	Hyp	Ref	Expression
1		fuconuniq2.c	. 2 ⊢ 𝐶 = (SetCat‘1o)
2		fuconuniq2.e	. 2 ⊢ 𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}
3		fuconuniq2.q	. 2 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
4		fuconuniq2.l	. 2 ⊢ 𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)
5		fuconuniq.a	. 2 ⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
6		fuconuniq.m	. 2 ⊢ 𝑀 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))
7		fuconuniq.s	. 2 ⊢ 𝑆 = ⟨( I ↾ 2o), 𝑀⟩
8		fuconuniq.r	. 2 ⊢ 𝑅 = (𝐸 FuncCat 𝐸)
9		fuconuniq.o	. 2 ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))
10		fuconuniq.k	. 2 ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝑆)
11		fuconuniqvals.d	. 2 ⊢ 𝐷 = (𝐸Δfunc𝐶)
12		fuconuniqvals.f	. 2 ⊢ 𝐹 = ((1st ‘𝐷)‘∅)
13		fuconuniqvals.g	. 2 ⊢ 𝐺 = ((1st ‘𝐷)‘1o)
14		fuconuniqvals.n	. 2 ⊢ 𝑁 = (1o × {∅})
15		eqid 2763	. 2 ⊢ ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶)) = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	fuconuniqvalslem 50988	1 ⊢ (((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = ∅ ∧ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = 1o)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∩ cin 4092  ∅c0 4281  ifcif 4485  {csn 4588  {ctp 4592  ⟨cop 4838   ↦ cmpt 5189   I cid 5549   × cxp 5653   ↾ cres 5657  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  1^st c1st 8023  2^nd c2nd 8024  1_oc1o 8497  2_oc2o 8498  3_oc3o 8499  ndxcnx 17400  Basecbs 17416  Hom chom 17468  compcco 17469  Idccid 17868  id_funccidfu 18059   ∘_func ccofu 18060   FuncCat cfuc 18149  SetCatcsetc 18279   ×_c cxpc 18371   curry_F ccurf 18413  Δ_funccdiag 18415   ∘_F cfuco 50643

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-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

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-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  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-tp 4593  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-riota 7400  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-1o 8504  df-2o 8505  df-3o 8506  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-cat 17871  df-cid 17872  df-func 18062  df-idfu 18063  df-cofu 18064  df-nat 18150  df-fuc 18151  df-setc 18280  df-xpc 18375  df-1stf 18376  df-curf 18417  df-diag 18419  df-fuco 50644  df-thinc 50758  df-termc 50813

This theorem is used by: (None)