lanval2 - Mathbox for Zhi Wang

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Theorem lanval2 50131

Description: The set of left Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by isup2 49698 and upciclem1 49670. (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 50129	. . . 4 ⊢ (𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋))
5	4	adantl 483	. . 3 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)) → 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋))
6		simpr 486	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋))
7		simpl 484	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → 𝐹 ∈ (𝐶 Func 𝐷))
8	2	fucbas 17925	. . . . . . . 8 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
9	8	uprcl 49688	. . . . . . 7 ⊢ (𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋) → (𝐾 ∈ (𝑅 Func 𝑆) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
10	9	simprd 497	. . . . . 6 ⊢ (𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
11	10	adantl 483	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → 𝑋 ∈ (𝐶 Func 𝐸))
12	3	eqcomi 2750	. . . . . 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 50123	. . . 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 2844	. . 3 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → 𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋))
16	5, 15	impbida 807	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ↔ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)))
17	16	eqrdv 2739	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 397 = wceq 1548 ∈ wcel 2121 ⟨cop 4564 (class class class)co 7360 Func cfunc 17816 FuncCat cfuc 17907 UP cup 49677 −∘_F cprcof 49877 Lan clan 50109

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-hom 17239 df-cco 17240 df-func 17820 df-fuc 17909 df-up 49678 df-lan 50111

This theorem is referenced by: cmdlan 50176

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