Theorem par2diag 50871

Description: Two constant functors into the concrete parallel-arrow category and a natural transformation whose component is the empty-set arrow. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
par2diag.c	⊢ 𝐶 = (SetCat‘1o)
par2diag.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)))⟩}
par2diag.l	⊢ 𝐿 = (𝐸Δfunc𝐶)
par2diag.f	⊢ 𝐹 = ((1st ‘𝐿)‘∅)
par2diag.g	⊢ 𝐺 = ((1st ‘𝐿)‘1o)
par2diag.a	⊢ 𝐴 = (1o × {∅})

Assertion

Ref	Expression
par2diag	⊢ ((𝐹 ∈ (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) ∧ (𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺) ∧ (𝐴‘∅) = ∅) ∧ (((1st ‘𝐹)‘∅) = ∅ ∧ ((1st ‘𝐺)‘∅) = 1o))

Distinct variable group:   𝑚,𝑛,𝑟,𝑠,𝑝,𝑞

Allowed substitution hints:   𝐴(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝐶(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝐸(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝐹(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝐺(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝐿(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)

Proof of Theorem par2diag

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		par2diag.l	. . . . 5 ⊢ 𝐿 = (𝐸Δfunc𝐶)
2		par2diag.e	. . . . . . . 8 ⊢ 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	2	par2catid 50858	. . . . . . 7 ⊢ (𝐸 ∈ Cat ∧ (Id‘𝐸) = (𝑦 ∈ 2o ↦ 2o))
4	3	simpli 489	. . . . . 6 ⊢ 𝐸 ∈ Cat
5	4	a1i 11	. . . . 5 ⊢ (⊤ → 𝐸 ∈ Cat)
6		par2diag.c	. . . . . . . 8 ⊢ 𝐶 = (SetCat‘1o)
7		setc1oterm 50717	. . . . . . . 8 ⊢ (SetCat‘1o) ∈ TermCat
8	6, 7	eqeltri 2856	. . . . . . 7 ⊢ 𝐶 ∈ TermCat
9	8	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐶 ∈ TermCat)
10	9	termccatd 50705	. . . . 5 ⊢ (⊤ → 𝐶 ∈ Cat)
11	2	par2str 50860	. . . . . 6 ⊢ (2o = (Base‘𝐸) ∧ (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅))) = (Hom ‘𝐸) ∧ (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠))) = (comp‘𝐸))
12	11	simp1i 1157	. . . . 5 ⊢ 2o = (Base‘𝐸)
13		0ex 5260	. . . . . . . 8 ⊢ ∅ ∈ V
14	13	prid1 4722	. . . . . . 7 ⊢ ∅ ∈ {∅, 1o}
15		df2o3 8486	. . . . . . 7 ⊢ 2o = {∅, 1o}
16	14, 15	eleqtrri 2859	. . . . . 6 ⊢ ∅ ∈ 2o
17	16	a1i 11	. . . . 5 ⊢ (⊤ → ∅ ∈ 2o)
18		par2diag.f	. . . . 5 ⊢ 𝐹 = ((1st ‘𝐿)‘∅)
19	1, 5, 10, 12, 17, 18	diag1cl 18418	. . . 4 ⊢ (⊤ → 𝐹 ∈ (𝐶 Func 𝐸))
20		1oelpr 8489	. . . . . . 7 ⊢ 1o ∈ {∅, 1o}
21	20, 15	eleqtrri 2859	. . . . . 6 ⊢ 1o ∈ 2o
22	21	a1i 11	. . . . 5 ⊢ (⊤ → 1o ∈ 2o)
23		par2diag.g	. . . . 5 ⊢ 𝐺 = ((1st ‘𝐿)‘1o)
24	1, 5, 10, 12, 22, 23	diag1cl 18418	. . . 4 ⊢ (⊤ → 𝐺 ∈ (𝐶 Func 𝐸))
25	19, 24	jca 521	. . 3 ⊢ (⊤ → (𝐹 ∈ (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)))
26	6	setc1obas 50718	. . . . . 6 ⊢ 1o = (Base‘𝐶)
27	11	simp2i 1158	. . . . . 6 ⊢ (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅))) = (Hom ‘𝐸)
28		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅))) = (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))
29	28	par2hom 50870	. . . . . . . . . 10 ⊢ ((∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))∅) = {2o} ∧ (1o(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o) = {2o} ∧ ((∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o) = 2o ∧ (1o(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))∅) = ∅))
30	29	simp3i 1159	. . . . . . . . 9 ⊢ ((∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o) = 2o ∧ (1o(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))∅) = ∅)
31	30	simpli 489	. . . . . . . 8 ⊢ (∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o) = 2o
32	16, 31	eleqtrri 2859	. . . . . . 7 ⊢ ∅ ∈ (∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o)
33	32	a1i 11	. . . . . 6 ⊢ (⊤ → ∅ ∈ (∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o))
34		eqid 2760	. . . . . 6 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
35	1, 12, 26, 27, 5, 10, 17, 22, 33, 34	diag2cl 18422	. . . . 5 ⊢ (⊤ → (1o × {∅}) ∈ (((1st ‘𝐿)‘∅)(𝐶 Nat 𝐸)((1st ‘𝐿)‘1o)))
36		par2diag.a	. . . . 5 ⊢ 𝐴 = (1o × {∅})
37	18, 23	oveq12i 7432	. . . . 5 ⊢ (𝐹(𝐶 Nat 𝐸)𝐺) = (((1st ‘𝐿)‘∅)(𝐶 Nat 𝐸)((1st ‘𝐿)‘1o))
38	35, 36, 37	3eltr4g 2877	. . . 4 ⊢ (⊤ → 𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))
39	36	fveq1i 6886	. . . . 5 ⊢ (𝐴‘∅) = ((1o × {∅})‘∅)
40		0lt1o 8514	. . . . . 6 ⊢ ∅ ∈ 1o
41	13	fvconst2 7210	. . . . . 6 ⊢ (∅ ∈ 1o → ((1o × {∅})‘∅) = ∅)
42	40, 41	ax-mp 5	. . . . 5 ⊢ ((1o × {∅})‘∅) = ∅
43	39, 42	eqtri 2783	. . . 4 ⊢ (𝐴‘∅) = ∅
44	38, 43	jctir 530	. . 3 ⊢ (⊤ → (𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺) ∧ (𝐴‘∅) = ∅))
45	40	a1i 11	. . . . 5 ⊢ (⊤ → ∅ ∈ 1o)
46	1, 5, 10, 12, 17, 18, 26, 45	diag11 18419	. . . 4 ⊢ (⊤ → ((1st ‘𝐹)‘∅) = ∅)
47	1, 5, 10, 12, 22, 23, 26, 45	diag11 18419	. . . 4 ⊢ (⊤ → ((1st ‘𝐺)‘∅) = 1o)
48	46, 47	jca 521	. . 3 ⊢ (⊤ → (((1st ‘𝐹)‘∅) = ∅ ∧ ((1st ‘𝐺)‘∅) = 1o))
49	25, 44, 48	3jca 1146	. 2 ⊢ (⊤ → ((𝐹 ∈ (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) ∧ (𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺) ∧ (𝐴‘∅) = ∅) ∧ (((1st ‘𝐹)‘∅) = ∅ ∧ ((1st ‘𝐺)‘∅) = 1o)))
50	49	mptru 1576	1 ⊢ ((𝐹 ∈ (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) ∧ (𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺) ∧ (𝐴‘∅) = ∅) ∧ (((1st ‘𝐹)‘∅) = ∅ ∧ ((1st ‘𝐺)‘∅) = 1o))

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144   ∩ cin 4089  ∅c0 4278  ifcif 4481  {csn 4583  {cpr 4585  {ctp 4587  ⟨cop 4589   ↦ cmpt 5185   × cxp 5648  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  1_oc1o 8471  2_oc2o 8472  3_oc3o 8473  ndxcnx 17373  Basecbs 17389  Hom chom 17441  compcco 17442  Catccat 17840  Idccid 17841   Func cfunc 18031   Nat cnat 18121  SetCatcsetc 18252  Δ_funccdiag 18388  TermCatctermc 50698

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-nat 18123  df-fuc 18124  df-setc 18253  df-xpc 18348  df-1stf 18349  df-curf 18390  df-diag 18392  df-thinc 50644  df-termc 50699

This theorem is used by:  fuconuniqlem  50872  fuconuniqvalslem  50874