Theorem cmdlan 50779

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 50641). (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 50763	. . 3 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
2	1	breqi 5109	. 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 50292	. . . . . 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 2761	. . . . . . 7 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
6	5	fucbas 18138	. . . . . 6 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
7	4, 6	uprcl3 50297	. . . . 5 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
8		eqid 2761	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
9	4, 8	uprcl4 50298	. . . . 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 50292	. . . . . 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 50297	. . . . 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 2761	. . . . . . . . . . 11 ⊢ ( 1 FuncCat 𝐶) = ( 1 FuncCat 𝐶)
17	16	fucbas 18138	. . . . . . . . . 10 ⊢ ( 1 Func 𝐶) = (Base‘( 1 FuncCat 𝐶))
18	12, 17	uprcl4 50298	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 ∈ ( 1 Func 𝐶))
19		relfunc 18037	. . . . . . . . 9 ⊢ Rel ( 1 Func 𝐶)
20	18, 19	oppfrcllem 50234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 ≠ ∅)
21	15, 20	eqnetrrd 3024	. . . . . . 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 50183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
29	28	funcrcl3 50187	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐶 ∈ Cat)
30		lmdran.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc 1 )
31	8, 26, 29, 30	diag1f1o 50641	. . . . . . . . 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 2863	. . . . 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 2761	. . . . . 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 2762	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺))
44	30, 39, 41, 42, 43	prcofdiag 50501	. . . . 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 50493	. . . . 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 50645	. . . . . 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 50329	. . . 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 2761	. . . . . 6 ⊢ (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺)
55	16, 5, 54	lanval2 50734	. . . . 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 5114	. . 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 2145   ≠ wne 2956   ∩ cin 3898  ∅c0 4279  {csn 4584  ⟨cop 4590   class class class wbr 5103  dom cdm 5651   ↾ cres 5653  Fun wfun 6532  ⟶wf 6534  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17387  Catccat 17838   Func cfunc 18029   Full cful 18079   Faith cfth 18080   FuncCat cfuc 18120  Δ_funccdiag 18386   UP cup 50280   −∘_F cprcof 50480  TermCatctermc 50579   Lan clan 50712   Colimit ccmd 50751

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-cat 17842  df-cid 17843  df-func 18033  df-cofu 18035  df-full 18081  df-fth 18082  df-nat 18121  df-fuc 18122  df-xpc 18346  df-1stf 18347  df-curf 18388  df-diag 18390  df-up 50281  df-swapf 50367  df-fuco 50424  df-prcof 50481  df-thinc 50525  df-termc 50580  df-lan 50714  df-cmd 50753

This theorem is used by: (None)