Theorem cmdlan 50997

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 50811). (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 50981	. . 3 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
2	1	breqi 5112	. 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 50443	. . . . . 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 2763	. . . . . . 7 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
6	5	fucbas 18153	. . . . . 6 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
7	4, 6	uprcl3 50448	. . . . 5 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
8		eqid 2763	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
9	4, 8	uprcl4 50449	. . . . 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 50443	. . . . . 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 50448	. . . . 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 2763	. . . . . . . . . . 11 ⊢ ( 1 FuncCat 𝐶) = ( 1 FuncCat 𝐶)
17	16	fucbas 18153	. . . . . . . . . 10 ⊢ ( 1 Func 𝐶) = (Base‘( 1 FuncCat 𝐶))
18	12, 17	uprcl4 50449	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 ∈ ( 1 Func 𝐶))
19		relfunc 18052	. . . . . . . . 9 ⊢ Rel ( 1 Func 𝐶)
20	18, 19	oppfrcllem 50385	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 ≠ ∅)
21	15, 20	eqnetrrd 3027	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → ((1st ‘𝐿)‘𝑋) ≠ ∅)
22		fvfundmfvn0 6936	. . . . . . . 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 50332	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
29	28	funcrcl3 50336	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐶 ∈ Cat)
30		lmdran.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc 1 )
31	8, 26, 29, 30	diag1f1o 50811	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
32		f1of 6835	. . . . . . . . 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 6731	. . . . . . 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 2866	. . . . 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 2763	. . . . . 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 2764	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺))
44	30, 39, 41, 42, 43	prcofdiag 50671	. . . . 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 50663	. . . . 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 50818	. . . . . 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 6843	. . . . . . 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 50480	. . . 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 2763	. . . . . 6 ⊢ (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺)
55	16, 5, 54	lanval2 50952	. . . . 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 5117	. . 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 1569   ∈ wcel 2146   ≠ wne 2959   ∩ cin 4092  ∅c0 4281  {csn 4588  ⟨cop 4838   class class class wbr 5106  dom cdm 5655   ↾ cres 5657  Fun wfun 6541  ⟶wf 6543  –onto→wfo 6545  –1-1-onto→wf1o 6546  ‘cfv 6547  (class class class)co 7430  1^st c1st 8009  2^nd c2nd 8010  Basecbs 17402  Catccat 17853   Func cfunc 18044   Full cful 18094   Faith cfth 18095   FuncCat cfuc 18135  Δ_funccdiag 18401   UP cup 50431   −∘_F cprcof 50650  TermCatctermc 50749   Lan clan 50930   Colimit ccmd 50969

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290

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 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-er 8731  df-map 8863  df-ixp 8940  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-5 12421  df-6 12422  df-7 12423  df-8 12424  df-9 12425  df-n0 12620  df-z 12707  df-dec 12828  df-uz 12979  df-fz 13653  df-struct 17340  df-slot 17375  df-ndx 17387  df-base 17403  df-hom 17467  df-cco 17468  df-cat 17857  df-cid 17858  df-func 18048  df-cofu 18050  df-full 18096  df-fth 18097  df-nat 18136  df-fuc 18137  df-xpc 18361  df-1stf 18362  df-curf 18403  df-diag 18405  df-up 50432  df-swapf 50524  df-fuco 50581  df-prcof 50651  df-thinc 50695  df-termc 50750  df-lan 50932  df-cmd 50971

This theorem is used by: (None)