islan - Mathbox for Zhi Wang

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Theorem islan 50288

Description: A left Kan extension is a universal pair. (Contributed by Zhi Wang, 3-Nov-2025.)

**Hypotheses**
Ref	Expression
islan.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
islan.s	⊢ 𝑆 = (𝐶 FuncCat 𝐸)
islan.k	⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘_F 𝐹)

**Assertion**
Ref	Expression
islan	⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐿 ∈ (𝐾(𝑅 UP 𝑆)𝑋))

**Proof of Theorem islan**
Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋))
2		islan.r	. . 3 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
3		islan.s	. . 3 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4		lanrcl 50284	. . . 4 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
5	4	simpld 499	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐹 ∈ (𝐶 Func 𝐷))
6	4	simprd 500	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
7		islan.k	. . . . 5 ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘_F 𝐹)
8	7	eqcomi 2778	. . . 4 ⊢ (⟨𝐷, 𝐸⟩ −∘_F 𝐹) = 𝐾
9	8	a1i 11	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (⟨𝐷, 𝐸⟩ −∘_F 𝐹) = 𝐾)
10	2, 3, 5, 6, 9	lanval 50282	. 2 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))
11	1, 10	eleqtrd 2871	1 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐿 ∈ (𝐾(𝑅 UP 𝑆)𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1567 ∈ wcel 2149 ⟨cop 4600 (class class class)co 7411 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: islan2 50289 lanval2 50290 lanrcl4 50297

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