islan - Mathbox for Zhi Wang

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Theorem islan 49812

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 22	. 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 49808	. . . 4 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
5	4	simpld 494	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐹 ∈ (𝐶 Func 𝐷))
6	4	simprd 495	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
7		islan.k	. . . . 5 ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘_F 𝐹)
8	7	eqcomi 2743	. . . 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 49806	. 2 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))
11	1, 10	eleqtrd 2836	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 1541 ∈ wcel 2113 ⟨cop 4584 (class class class)co 7356 Func cfunc 17776 FuncCat cfuc 17867 UP cup 49360 −∘_F cprcof 49560 Lan clan 49792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-func 17780 df-lan 49794

This theorem is referenced by: islan2 49813 lanval2 49814 lanrcl4 49821

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