Theorem lanval 50282

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 49739	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
5	4	funcrcl2 49742	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	4	funcrcl3 49743	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		lanval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
8	7	func1st2nd 49739	. . . 4 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
9	8	funcrcl3 49743	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
10	1, 2, 5, 6, 9	lanfval 50276	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
11		simprl 782	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹)
12	11	oveq2d 7427	. . . 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 485	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝐾)
15	12, 14	eqtrd 2804	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝑓) = 𝐾)
16		simprr 784	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋)
17	15, 16	oveq12d 7429	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥) = (𝐾(𝑅 UP 𝑆)𝑋))
18		ovexd 7446	. 2 ⊢ (𝜑 → (𝐾(𝑅 UP 𝑆)𝑋) ∈ V)
19	10, 17, 3, 7, 18	ovmpod 7563	1 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  2^nd c2nd 7985  Catccat 17720   Func cfunc 17911   FuncCat cfuc 18002   UP cup 49836   −∘_F cprcof 50036   Lan clan 50268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-func 17915  df-lan 50270

This theorem is referenced by:  rellan  50286  islan  50288  lanval2  50290  lanup  50304