Theorem cmdlan 50509

Description: To each colimit of a diagram there is a corresponding left Kan extention of the diagram along a functor to a terminal category. The morphism parts coincide, while the object parts are one-to-one correspondent (diag1f1o 50371). (Contributed by Zhi Wang, 26-Nov-2025.)

Hypotheses

Ref	Expression
lmdran.1	⊢ (𝜑 → 1 ∈ TermCat)
lmdran.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))
lmdran.l	⊢ 𝐿 = (𝐶Δfunc 1 )
lmdran.y	⊢ (𝜑 → 𝑌 = ((1st ‘𝐿)‘𝑋))

Assertion

Ref	Expression
cmdlan	⊢ (𝜑 → (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹)𝑀))

Proof of Theorem cmdlan

Step	Hyp	Ref	Expression
1		cmdfval2 50493	. . 3 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
2	1	breqi 5117	. 2 ⊢ (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑀 ↔ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀)
3		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀)
4	3	up1st2nd 50022	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑋(⟨(1st ‘(𝐶Δfunc𝐷)), (2nd ‘(𝐶Δfunc𝐷))⟩(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀)
5		eqid 2765	. . . . . . 7 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
6	5	fucbas 18044	. . . . . 6 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
7	4, 6	uprcl3 50027	. . . . 5 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
8		eqid 2765	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
9	4, 8	uprcl4 50028	. . . . 5 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑋 ∈ (Base‘𝐶))
10	7, 9	jca 521	. . . 4 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶)))
11		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀)
12	11	up1st2nd 50022	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌(⟨(1st ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺)), (2nd ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))⟩(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀)
13	12, 6	uprcl3 50027	. . . . 5 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
14		lmdran.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 = ((1st ‘𝐿)‘𝑋))
15	14	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 = ((1st ‘𝐿)‘𝑋))
16		eqid 2765	. . . . . . . . . . 11 ⊢ ( 1 FuncCat 𝐶) = ( 1 FuncCat 𝐶)
17	16	fucbas 18044	. . . . . . . . . 10 ⊢ ( 1 Func 𝐶) = (Base‘( 1 FuncCat 𝐶))
18	12, 17	uprcl4 50028	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 ∈ ( 1 Func 𝐶))
19		relfunc 17943	. . . . . . . . 9 ⊢ Rel ( 1 Func 𝐶)
20	18, 19	oppfrcllem 49964	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 ≠ ∅)
21	15, 20	eqnetrrd 3028	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → ((1st ‘𝐿)‘𝑋) ≠ ∅)
22		fvfundmfvn0 6925	. . . . . . . 8 ⊢ (((1st ‘𝐿)‘𝑋) ≠ ∅ → (𝑋 ∈ dom (1st ‘𝐿) ∧ Fun ((1st ‘𝐿) ↾ {𝑋})))
23	22	simpld 500	. . . . . . 7 ⊢ (((1st ‘𝐿)‘𝑋) ≠ ∅ → 𝑋 ∈ dom (1st ‘𝐿))
24	21, 23	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑋 ∈ dom (1st ‘𝐿))
25		lmdran.1	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ TermCat)
26	25	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 1 ∈ TermCat)
27		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐹 ∈ (𝐷 Func 𝐶))
28	27	func1st2nd 49913	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
29	28	funcrcl3 49917	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐶 ∈ Cat)
30		lmdran.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc 1 )
31	8, 26, 29, 30	diag1f1o 50371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
32		f1of 6824	. . . . . . . . 9 ⊢ ((1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶) → (1st ‘𝐿):(Base‘𝐶)⟶( 1 Func 𝐶))
33	31, 32	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)⟶( 1 Func 𝐶))
34	33	fdmd 6720	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → dom (1st ‘𝐿) = (Base‘𝐶))
35	13, 34	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → dom (1st ‘𝐿) = (Base‘𝐶))
36	24, 35	eleqtrd 2867	. . . . 5 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑋 ∈ (Base‘𝐶))
37	13, 36	jca 521	. . . 4 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶)))
38	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑌 = ((1st ‘𝐿)‘𝑋))
39		eqid 2765	. . . . . 6 ⊢ (𝐶Δfunc𝐷) = (𝐶Δfunc𝐷)
40		lmdran.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))
41	40	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 1 ))
42	29	adantrr 730	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
43		eqidd 2766	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺))
44	30, 39, 41, 42, 43	prcofdiag 50231	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((⟨ 1 , 𝐶⟩ −∘F 𝐺) ∘func 𝐿) = (𝐶Δfunc𝐷))
45		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑋 ∈ (Base‘𝐶))
46	16, 42, 5, 41	prcoffunca 50223	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) ∈ (( 1 FuncCat 𝐶) Func (𝐷 FuncCat 𝐶)))
47	29, 26, 16, 30	diagffth 50375	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐿 ∈ ((𝐶 Full ( 1 FuncCat 𝐶)) ∩ (𝐶 Faith ( 1 FuncCat 𝐶))))
48	47	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐿 ∈ ((𝐶 Full ( 1 FuncCat 𝐶)) ∩ (𝐶 Faith ( 1 FuncCat 𝐶))))
49		f1ofo 6832	. . . . . . 7 ⊢ ((1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶) → (1st ‘𝐿):(Base‘𝐶)–onto→( 1 Func 𝐶))
50	31, 49	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)–onto→( 1 Func 𝐶))
51	50	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘𝐿):(Base‘𝐶)–onto→( 1 Func 𝐶))
52	8, 17, 38, 44, 45, 46, 48, 51	uptr2a 50059	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀 ↔ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀))
53	10, 37, 52	bibiad 853	. . 3 ⊢ (𝜑 → (𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀 ↔ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀))
54		eqid 2765	. . . . . 6 ⊢ (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺)
55	16, 5, 54	lanval2 50464	. . . . 5 ⊢ (𝐺 ∈ (𝐷 Func 1 ) → (𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹) = ((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹))
56	40, 55	syl 18	. . . 4 ⊢ (𝜑 → (𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹) = ((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹))
57	56	breqd 5122	. . 3 ⊢ (𝜑 → (𝑌(𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹)𝑀 ↔ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀))
58	53, 57	bitr4d 285	. 2 ⊢ (𝜑 → (𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹)𝑀))
59	2, 58	bitrid 286	1 ⊢ (𝜑 → (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹)𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ∩ cin 3905  ∅c0 4286  {csn 4591  ⟨cop 4597   class class class wbr 5111  dom cdm 5663   ↾ cres 5665  Fun wfun 6534  ⟶wf 6536  –onto→wfo 6538  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991  Basecbs 17293  Catccat 17744   Func cfunc 17935   Full cful 17985   Faith cfth 17986   FuncCat cfuc 18026  Δ_funccdiag 18292   UP cup 50010   −∘_F cprcof 50210  TermCatctermc 50309   Lan clan 50442   Colimit ccmd 50481

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17231  df-slot 17266  df-ndx 17278  df-base 17294  df-hom 17358  df-cco 17359  df-cat 17748  df-cid 17749  df-func 17939  df-cofu 17941  df-full 17987  df-fth 17988  df-nat 18027  df-fuc 18028  df-xpc 18252  df-1stf 18253  df-curf 18294  df-diag 18296  df-up 50011  df-swapf 50097  df-fuco 50154  df-prcof 50211  df-thinc 50255  df-termc 50310  df-lan 50444  df-cmd 50483

This theorem is used by: (None)