lanval2 - Mathbox for Zhi Wang

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Theorem lanval2 49622

Description: The set of left Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by isup2 49189 and upciclem1 49161. (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
lanval2	⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))

Proof of Theorem lanval2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		islan.r	. . . . 5 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
2		islan.s	. . . . 5 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
3		islan.k	. . . . 5 ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘_F 𝐹)
4	1, 2, 3	islan 49620	. . . 4 ⊢ (𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋))
5	4	adantl 481	. . 3 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)) → 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋))
6		simpr 484	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋))
7		simpl 482	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → 𝐹 ∈ (𝐶 Func 𝐷))
8	2	fucbas 17870	. . . . . . . 8 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
9	8	uprcl 49179	. . . . . . 7 ⊢ (𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋) → (𝐾 ∈ (𝑅 Func 𝑆) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
10	9	simprd 495	. . . . . 6 ⊢ (𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
11	10	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → 𝑋 ∈ (𝐶 Func 𝐸))
12	3	eqcomi 2738	. . . . . 6 ⊢ (⟨𝐷, 𝐸⟩ −∘_F 𝐹) = 𝐾
13	12	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → (⟨𝐷, 𝐸⟩ −∘_F 𝐹) = 𝐾)
14	1, 2, 7, 11, 13	lanval 49614	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))
15	6, 14	eleqtrrd 2831	. . 3 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → 𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋))
16	5, 15	impbida 800	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ↔ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)))
17	16	eqrdv 2727	1 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⟨cop 4583 (class class class)co 7349 Func cfunc 17761 FuncCat cfuc 17852 UP cup 49168 −∘_F cprcof 49368 Lan clan 49600

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-fz 13411 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-hom 17185 df-cco 17186 df-func 17765 df-fuc 17854 df-up 49169 df-lan 49602

This theorem is referenced by: cmdlan 49667

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