lanval2 - Mathbox for Zhi Wang

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Theorem lanval2 50118

Description: The set of left Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by isup2 49685 and upciclem1 49657. (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 50116	. . . 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 17925	. . . . . . . 8 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
9	8	uprcl 49675	. . . . . . 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 2746	. . . . . 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 50110	. . . 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 2840	. . 3 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)) → 𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋))
16	5, 15	impbida 801	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝑥 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ↔ 𝑥 ∈ (𝐾(𝑅 UP 𝑆)𝑋)))
17	16	eqrdv 2735	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 1542 ∈ wcel 2114 ⟨cop 4574 (class class class)co 7362 Func cfunc 17816 FuncCat cfuc 17907 UP cup 49664 −∘_F cprcof 49864 Lan clan 50096

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 5304 ax-pr 5372 ax-un 7684 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 207 df-an 396 df-or 849 df-3or 1088 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-nel 3038 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-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 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 49665 df-lan 50098

This theorem is referenced by: cmdlan 50163

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