fuconuniq - Mathbox for Zhi Wang

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Theorem fuconuniq 50873

Description: The arrow map 𝐴 fixes the identity arrow 2_o and exchanges the two nonidentity parallel arrows ∅ and 1_o. It gives a concrete alternative to functor composition with the same object function and a different morphism function. All hypotheses name explicit constructed sets; no category or functor existence is assumed. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

**Hypotheses**
Ref	Expression
fuconuniq2.c	⊢ 𝐶 = (SetCat‘1_o)
fuconuniq2.e	⊢ 𝐸 = {⟨(Base‘ndx), 2_o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2_o, 𝑛 ∈ 2_o ↦ if(𝑚 = 𝑛, {2_o}, if((𝑚 = ∅ ∧ 𝑛 = 1_o), 2_o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2_o × 2_o), 𝑞 ∈ 2_o ↦ (𝑟 ∈ 3_o, 𝑠 ∈ 3_o ↦ (𝑟 ∩ 𝑠)))⟩}
fuconuniq2.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fuconuniq2.l	⊢ 𝐿 = (⟨𝐶, 𝐶⟩ ∘_F 𝐸)
fuconuniq.a	⊢ 𝐴 = (𝑏 ∈ 3_o ↦ if(𝑏 = 2_o, 2_o, if(𝑏 = ∅, 1_o, ∅)))
fuconuniq.m	⊢ 𝑀 = (𝑢 ∈ 2_o, 𝑣 ∈ 2_o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2_o, 𝑛 ∈ 2_o ↦ if(𝑚 = 𝑛, {2_o}, if((𝑚 = ∅ ∧ 𝑛 = 1_o), 2_o, ∅)))𝑣) ↦ (𝐴‘𝑎)))
fuconuniq.s	⊢ 𝑆 = ⟨( I ↾ 2_o), 𝑀⟩
fuconuniq.r	⊢ 𝑅 = (𝐸 FuncCat 𝐸)
fuconuniq.o	⊢ ⚬ = (⟨𝑅, 𝑄⟩ curry_F (⟨𝐶, 𝐸⟩ ∘_F 𝐸))
fuconuniq.k	⊢ 𝐾 = ((1^st ‘ ⚬ )‘𝑆)
fuconuniq.t	⊢ 𝑇 = ((𝐶 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐶))

**Assertion**
Ref	Expression
fuconuniq	⊢ ((1^st ‘𝐿)(𝑇 Func 𝑄)(2^nd ‘(𝐾 ∘_func 𝐿)) ∧ (2^nd ‘(𝐾 ∘_func 𝐿)) ≠ (2^nd ‘𝐿))

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) 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) 𝑇(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏) 𝐸(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) ⚬ (𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)

**Proof of Theorem fuconuniq**
Step	Hyp	Ref	Expression
1		fuconuniq2.c	. 2 ⊢ 𝐶 = (SetCat‘1_o)
2		fuconuniq2.e	. 2 ⊢ 𝐸 = {⟨(Base‘ndx), 2_o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2_o, 𝑛 ∈ 2_o ↦ if(𝑚 = 𝑛, {2_o}, if((𝑚 = ∅ ∧ 𝑛 = 1_o), 2_o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2_o × 2_o), 𝑞 ∈ 2_o ↦ (𝑟 ∈ 3_o, 𝑠 ∈ 3_o ↦ (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 ∈ 3_o ↦ if(𝑏 = 2_o, 2_o, if(𝑏 = ∅, 1_o, ∅)))
6		fuconuniq.m	. 2 ⊢ 𝑀 = (𝑢 ∈ 2_o, 𝑣 ∈ 2_o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2_o, 𝑛 ∈ 2_o ↦ if(𝑚 = 𝑛, {2_o}, if((𝑚 = ∅ ∧ 𝑛 = 1_o), 2_o, ∅)))𝑣) ↦ (𝐴‘𝑎)))
7		fuconuniq.s	. 2 ⊢ 𝑆 = ⟨( I ↾ 2_o), 𝑀⟩
8		fuconuniq.r	. 2 ⊢ 𝑅 = (𝐸 FuncCat 𝐸)
9		fuconuniq.o	. 2 ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curry_F (⟨𝐶, 𝐸⟩ ∘_F 𝐸))
10		fuconuniq.k	. 2 ⊢ 𝐾 = ((1^st ‘ ⚬ )‘𝑆)
11		fuconuniq.t	. 2 ⊢ 𝑇 = ((𝐶 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐶))
12		eqid 2760	. 2 ⊢ (𝐸Δ_func𝐶) = (𝐸Δ_func𝐶)
13		eqid 2760	. 2 ⊢ ((1^st ‘(𝐸Δ_func𝐶))‘∅) = ((1^st ‘(𝐸Δ_func𝐶))‘∅)
14		eqid 2760	. 2 ⊢ ((1^st ‘(𝐸Δ_func𝐶))‘1_o) = ((1^st ‘(𝐸Δ_func𝐶))‘1_o)
15		eqid 2760	. 2 ⊢ (1_o × {∅}) = (1_o × {∅})
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	fuconuniqlem 50872	1 ⊢ ((1^st ‘𝐿)(𝑇 Func 𝑄)(2^nd ‘(𝐾 ∘_func 𝐿)) ∧ (2^nd ‘(𝐾 ∘_func 𝐿)) ≠ (2^nd ‘𝐿))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1569 ≠ 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 Δ_funccdiag 18388 ∘_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: fuconuniq2 50876

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