Theorem lanval 51133

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 50521	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
5	4	funcrcl2 50524	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	4	funcrcl3 50525	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		lanval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
8	7	func1st2nd 50521	. . . 4 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
9	8	funcrcl3 50525	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
10	1, 2, 5, 6, 9	lanfval 51127	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
11		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹)
12	11	oveq2d 7467	. . . 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 486	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝐾)
15	12, 14	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝑓) = 𝐾)
16		simprr 785	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋)
17	15, 16	oveq12d 7469	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥) = (𝐾(𝑅 UP 𝑆)𝑋))
18		ovexd 7486	. 2 ⊢ (𝜑 → (𝐾(𝑅 UP 𝑆)𝑋) ∈ V)
19	10, 17, 3, 7, 18	ovmpod 7604	1 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4840  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  Catccat 17888   Func cfunc 18079   FuncCat cfuc 18170   UP cup 50620   −∘_F cprcof 50839   Lan clan 51119

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  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-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-func 18083  df-lan 51121

This theorem is used by:  rellan  51137  islan  51139  lanval2  51141  lanup  51155