Theorem lanval 50109

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 49566	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
5	4	funcrcl2 49569	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	4	funcrcl3 49570	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		lanval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
8	7	func1st2nd 49566	. . . 4 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
9	8	funcrcl3 49570	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
10	1, 2, 5, 6, 9	lanfval 50103	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
11		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹)
12	11	oveq2d 7377	. . . 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 2772	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝑓) = 𝐾)
16		simprr 773	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋)
17	15, 16	oveq12d 7379	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥) = (𝐾(𝑅 UP 𝑆)𝑋))
18		ovexd 7396	. 2 ⊢ (𝜑 → (𝐾(𝑅 UP 𝑆)𝑋) ∈ V)
19	10, 17, 3, 7, 18	ovmpod 7513	1 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574  ‘cfv 6493  (class class class)co 7361  1^st c1st 7934  2^nd c2nd 7935  Catccat 17624   Func cfunc 17815   FuncCat cfuc 17906   UP cup 49663   −∘_F cprcof 49863   Lan clan 50095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-func 17819  df-lan 50097

This theorem is referenced by:  rellan  50113  islan  50115  lanval2  50117  lanup  50131