Theorem cmdlan 49654

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 49516). (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 49638	. . 3 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
2	1	breqi 5108	. 2 ⊢ (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑀 ↔ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀)
3		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀)
4	3	up1st2nd 49167	. . . . . 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 2729	. . . . . . 7 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
6	5	fucbas 17905	. . . . . 6 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
7	4, 6	uprcl3 49172	. . . . 5 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
8		eqid 2729	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
9	4, 8	uprcl4 49173	. . . . 5 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑋 ∈ (Base‘𝐶))
10	7, 9	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶)))
11		simpr 484	. . . . . . 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 49167	. . . . . 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 49172	. . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 = ((1st ‘𝐿)‘𝑋))
16		eqid 2729	. . . . . . . . . . 11 ⊢ ( 1 FuncCat 𝐶) = ( 1 FuncCat 𝐶)
17	16	fucbas 17905	. . . . . . . . . 10 ⊢ ( 1 Func 𝐶) = (Base‘( 1 FuncCat 𝐶))
18	12, 17	uprcl4 49173	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 ∈ ( 1 Func 𝐶))
19		relfunc 17804	. . . . . . . . 9 ⊢ Rel ( 1 Func 𝐶)
20	18, 19	oppfrcllem 49109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑌 ≠ ∅)
21	15, 20	eqnetrrd 2993	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → ((1st ‘𝐿)‘𝑋) ≠ ∅)
22		fvfundmfvn0 6883	. . . . . . . 8 ⊢ (((1st ‘𝐿)‘𝑋) ≠ ∅ → (𝑋 ∈ dom (1st ‘𝐿) ∧ Fun ((1st ‘𝐿) ↾ {𝑋})))
23	22	simpld 494	. . . . . . 7 ⊢ (((1st ‘𝐿)‘𝑋) ≠ ∅ → 𝑋 ∈ dom (1st ‘𝐿))
24	21, 23	syl 17	. . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 1 ∈ TermCat)
27		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐹 ∈ (𝐷 Func 𝐶))
28	27	func1st2nd 49058	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
29	28	funcrcl3 49062	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐶 ∈ Cat)
30		lmdran.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc 1 )
31	8, 26, 29, 30	diag1f1o 49516	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
32		f1of 6782	. . . . . . . . 9 ⊢ ((1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶) → (1st ‘𝐿):(Base‘𝐶)⟶( 1 Func 𝐶))
33	31, 32	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)⟶( 1 Func 𝐶))
34	33	fdmd 6680	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → dom (1st ‘𝐿) = (Base‘𝐶))
35	13, 34	syldan 591	. . . . . 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 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑌((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹)𝑀) → 𝑋 ∈ (Base‘𝐶))
37	13, 36	jca 511	. . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑌 = ((1st ‘𝐿)‘𝑋))
39		eqid 2729	. . . . . 6 ⊢ (𝐶Δfunc𝐷) = (𝐶Δfunc𝐷)
40		lmdran.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))
41	40	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 1 ))
42	29	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
43		eqidd 2730	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺))
44	30, 39, 41, 42, 43	prcofdiag 49376	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((⟨ 1 , 𝐶⟩ −∘F 𝐺) ∘func 𝐿) = (𝐶Δfunc𝐷))
45		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑋 ∈ (Base‘𝐶))
46	16, 42, 5, 41	prcoffunca 49368	. . . . 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 49520	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐿 ∈ ((𝐶 Full ( 1 FuncCat 𝐶)) ∩ (𝐶 Faith ( 1 FuncCat 𝐶))))
48	47	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐿 ∈ ((𝐶 Full ( 1 FuncCat 𝐶)) ∩ (𝐶 Faith ( 1 FuncCat 𝐶))))
49		f1ofo 6789	. . . . . . 7 ⊢ ((1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶) → (1st ‘𝐿):(Base‘𝐶)–onto→( 1 Func 𝐶))
50	31, 49	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)–onto→( 1 Func 𝐶))
51	50	adantrr 717	. . . . 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 49204	. . . 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 839	. . 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 2729	. . . . . 6 ⊢ (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺)
55	16, 5, 54	lanval2 49609	. . . . 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 17	. . . 4 ⊢ (𝜑 → (𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹) = ((⟨ 1 , 𝐶⟩ −∘F 𝐺)(( 1 FuncCat 𝐶) UP (𝐷 FuncCat 𝐶))𝐹))
57	56	breqd 5113	. . 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 282	. 2 ⊢ (𝜑 → (𝑋((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹)𝑀))
59	2, 58	bitrid 283	1 ⊢ (𝜑 → (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹)𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∩ cin 3910  ∅c0 4292  {csn 4585  ⟨cop 4591   class class class wbr 5102  dom cdm 5631   ↾ cres 5633  Fun wfun 6493  ⟶wf 6495  –onto→wfo 6497  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155  Catccat 17605   Func cfunc 17796   Full cful 17846   Faith cfth 17847   FuncCat cfuc 17887  Δ_funccdiag 18153   UP cup 49155   −∘_F cprcof 49355  TermCatctermc 49454   Lan clan 49587   Colimit ccmd 49626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-cat 17609  df-cid 17610  df-func 17800  df-cofu 17802  df-full 17848  df-fth 17849  df-nat 17888  df-fuc 17889  df-xpc 18113  df-1stf 18114  df-curf 18155  df-diag 18157  df-up 49156  df-swapf 49242  df-fuco 49299  df-prcof 49356  df-thinc 49400  df-termc 49455  df-lan 49589  df-cmd 49628

This theorem is referenced by: (None)