Theorem lanval 49614

Description: Value of the set of left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)

Hypotheses

Ref	Expression
lanval.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
lanval.s	⊢ 𝑆 = (𝐶 FuncCat 𝐸)
lanval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
lanval.x	⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
lanval.k	⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝐾)

Assertion

Ref	Expression
lanval	⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))

Proof of Theorem lanval

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lanval.r	. . 3 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
2		lanval.s	. . 3 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
3		lanval.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4	3	func1st2nd 49071	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
5	4	funcrcl2 49074	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	4	funcrcl3 49075	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		lanval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
8	7	func1st2nd 49071	. . . 4 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
9	8	funcrcl3 49075	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
10	1, 2, 5, 6, 9	lanfval 49608	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
11		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹)
12	11	oveq2d 7365	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝑓) = (⟨𝐷, 𝐸⟩ −∘F 𝐹))
13		lanval.k	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝐾)
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝐾)
15	12, 14	eqtrd 2764	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝑓) = 𝐾)
16		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋)
17	15, 16	oveq12d 7367	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥) = (𝐾(𝑅 UP 𝑆)𝑋))
18		ovexd 7384	. 2 ⊢ (𝜑 → (𝐾(𝑅 UP 𝑆)𝑋) ∈ V)
19	10, 17, 3, 7, 18	ovmpod 7501	1 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  Catccat 17570   Func cfunc 17761   FuncCat cfuc 17852   UP cup 49168   −∘_F cprcof 49368   Lan clan 49600

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-func 17765  df-lan 49602

This theorem is referenced by:  rellan  49618  islan  49620  lanval2  49622  lanup  49636