Theorem fuconuniq2 50990

Description: The concrete parallel-arrow category supplies a morphism function different from canonical functor composition, with its object function fixed exactly. See par2cat 50973 and par2hom 50984 for the category and its four hom-sets. (Contributed by Zhi Wang, 14-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)
fuconuniq2.t	⊢ 𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))

Assertion

Ref	Expression
fuconuniq2	⊢ ∃𝑧((1st ‘𝐿)(𝑇 Func 𝑄)𝑧 ∧ 𝑧 ≠ (2nd ‘𝐿))

Distinct variable groups:   𝑚,𝑛,𝑟,𝑠,𝑝,𝑞,𝑧   𝑧,𝐶   𝑧,𝐸   𝑧,𝑄   𝑧,𝐿   𝑧,𝑇

Allowed substitution hints:   𝐶(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝑄(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝑇(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝐸(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝐿(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)

Proof of Theorem fuconuniq2

Dummy variables 𝑢 𝑣 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6921	. 2 ⊢ (2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿)) ∈ V
2		breq2 5110	. . 3 ⊢ (𝑧 = (2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿)) → ((1st ‘𝐿)(𝑇 Func 𝑄)𝑧 ↔ (1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿))))
3		neeq1 3021	. . 3 ⊢ (𝑧 = (2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿)) → (𝑧 ≠ (2nd ‘𝐿) ↔ (2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿)) ≠ (2nd ‘𝐿)))
4	2, 3	anbi12d 644	. 2 ⊢ (𝑧 = (2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿)) → (((1st ‘𝐿)(𝑇 Func 𝑄)𝑧 ∧ 𝑧 ≠ (2nd ‘𝐿)) ↔ ((1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿)) ∧ (2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿)) ≠ (2nd ‘𝐿))))
5		fuconuniq2.c	. . 3 ⊢ 𝐶 = (SetCat‘1o)
6		fuconuniq2.e	. . 3 ⊢ 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)))⟩}
7		fuconuniq2.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
8		fuconuniq2.l	. . 3 ⊢ 𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)
9		eqid 2763	. . 3 ⊢ (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅))) = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
10		eqid 2763	. . 3 ⊢ (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎))) = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))
11		eqid 2763	. . 3 ⊢ ⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩ = ⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩
12		eqid 2763	. . 3 ⊢ (𝐸 FuncCat 𝐸) = (𝐸 FuncCat 𝐸)
13		eqid 2763	. . 3 ⊢ (⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)) = (⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))
14		eqid 2763	. . 3 ⊢ ((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) = ((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩)
15		fuconuniq2.t	. . 3 ⊢ 𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))
16	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	fuconuniq 50987	. 2 ⊢ ((1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿)) ∧ (2nd ‘(((1st ‘(⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)))‘⟨( I ↾ 2o), (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ ((𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))‘𝑎)))⟩) ∘func 𝐿)) ≠ (2nd ‘𝐿))
17	1, 4, 16	ceqsexv2d 3503	1 ⊢ ∃𝑧((1st ‘𝐿)(𝑇 Func 𝑄)𝑧 ∧ 𝑧 ≠ (2nd ‘𝐿))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ≠ wne 2959   ∩ cin 4092  ∅c0 4281  ifcif 4485  {csn 4588  {ctp 4592  ⟨cop 4838   class class class wbr 5106   ↦ 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   Func cfunc 18058   ∘_func ccofu 18060   FuncCat cfuc 18149  SetCatcsetc 18279   ×_c cxpc 18371   curry_F ccurf 18413   ∘_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:  fuconuniqex  50991