Theorem fuconuniq2 50876

Description: The concrete parallel-arrow category supplies a morphism function different from canonical functor composition, with its object function fixed exactly. See par2cat 50859 and par2hom 50870 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 6898	. 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 5106	. . 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 3017	. . 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 2760	. . 3 ⊢ (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅))) = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
10		eqid 2760	. . 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 2760	. . 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 2760	. . 3 ⊢ (𝐸 FuncCat 𝐸) = (𝐸 FuncCat 𝐸)
13		eqid 2760	. . 3 ⊢ (⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸)) = (⟨(𝐸 FuncCat 𝐸), 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))
14		eqid 2760	. . 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 50873	. 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 3499	1 ⊢ ∃𝑧((1st ‘𝐿)(𝑇 Func 𝑄)𝑧 ∧ 𝑧 ≠ (2nd ‘𝐿))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ∃wex 1811   ≠ wne 2955   ∩ cin 4089  ∅c0 4278  ifcif 4481  {csn 4583  {ctp 4587  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185   I cid 5544   × cxp 5648   ↾ cres 5652  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  1_oc1o 8471  2_oc2o 8472  3_oc3o 8473  ndxcnx 17373  Basecbs 17389  Hom chom 17441  compcco 17442   Func cfunc 18031   ∘_func ccofu 18033   FuncCat cfuc 18122  SetCatcsetc 18252   ×_c cxpc 18344   curry_F ccurf 18386   ∘_F cfuco 50529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11258  ax-resscn 11259  ax-1cn 11260  ax-icn 11261  ax-addcl 11262  ax-addrcl 11263  ax-mulcl 11264  ax-mulrcl 11265  ax-mulcom 11266  ax-addass 11267  ax-mulass 11268  ax-distr 11269  ax-i2m1 11270  ax-1ne0 11271  ax-1rid 11272  ax-rnegex 11273  ax-rrecex 11274  ax-cnre 11275  ax-pre-lttri 11276  ax-pre-lttrn 11277  ax-pre-ltadd 11278  ax-pre-mulgt0 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8301  df-wrecs 8332  df-recs 8381  df-rdg 8420  df-1o 8478  df-2o 8479  df-3o 8480  df-er 8719  df-map 8851  df-ixp 8928  df-en 8976  df-dom 8977  df-sdom 8978  df-fin 8979  df-pnf 11347  df-mnf 11348  df-xr 11349  df-ltxr 11350  df-le 11351  df-sub 11545  df-neg 11546  df-nn 12338  df-2 12407  df-3 12408  df-4 12409  df-5 12410  df-6 12411  df-7 12412  df-8 12413  df-9 12414  df-n0 12609  df-z 12696  df-dec 12817  df-uz 12968  df-fz 13642  df-struct 17327  df-slot 17362  df-ndx 17374  df-base 17390  df-hom 17454  df-cco 17455  df-cat 17844  df-cid 17845  df-func 18035  df-idfu 18036  df-cofu 18037  df-nat 18123  df-fuc 18124  df-setc 18253  df-xpc 18348  df-1stf 18349  df-curf 18390  df-diag 18392  df-fuco 50530  df-thinc 50644  df-termc 50699

This theorem is used by:  fuconuniqex  50877