Theorem lanrcl 50284

Description: Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)

Assertion

Ref	Expression
lanrcl	⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))

Proof of Theorem lanrcl

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋))
2		ne0i 4302	. . . . 5 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅)
3		eqid 2769	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4		eqid 2769	. . . . . 6 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
5		df-ov 7414	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ Lan 𝐸) = ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩)
6	5	eqeq1i 2774	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ ↔ ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅)
7		oveq 7417	. . . . . . . . . 10 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7448	. . . . . . . . . 10 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2820	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ∅)
10	6, 9	sylbir 238	. . . . . . . 8 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ∅)
11	10	necon3i 2996	. . . . . . 7 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅)
12		fvfundmfvn0 6922	. . . . . . . . 9 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Lan ∧ Fun ( Lan ↾ {⟨⟨𝐶, 𝐷⟩, 𝐸⟩})))
13	12	simpld 499	. . . . . . . 8 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Lan )
14		lanfn 50272	. . . . . . . . 9 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6640	. . . . . . . 8 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2879	. . . . . . 7 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5704	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → ⟨𝐶, 𝐷⟩ ∈ (V × V))
18		opelxp1 5704	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐶 ∈ V)
19	11, 16, 17, 18	4syl 20	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → 𝐶 ∈ V)
20		opelxp2 5705	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐷 ∈ V)
21	11, 16, 17, 20	4syl 20	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → 𝐷 ∈ V)
22		opelxp2 5705	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
23	11, 16, 22	3syl 19	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → 𝐸 ∈ V)
24	3, 4, 19, 21, 23	lanfval 50276	. . . . 5 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)((𝐷 FuncCat 𝐸) UP (𝐶 FuncCat 𝐸))𝑥)))
25	2, 24	syl 18	. . . 4 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)((𝐷 FuncCat 𝐸) UP (𝐶 FuncCat 𝐸))𝑥)))
26	25	oveqd 7428	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐹(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)((𝐷 FuncCat 𝐸) UP (𝐶 FuncCat 𝐸))𝑥))𝑋))
27	1, 26	eleqtrd 2871	. 2 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐿 ∈ (𝐹(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)((𝐷 FuncCat 𝐸) UP (𝐶 FuncCat 𝐸))𝑥))𝑋))
28		eqid 2769	. . 3 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)((𝐷 FuncCat 𝐸) UP (𝐶 FuncCat 𝐸))𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)((𝐷 FuncCat 𝐸) UP (𝐶 FuncCat 𝐸))𝑥))
29	28	elmpocl 7652	. 2 ⊢ (𝐿 ∈ (𝐹(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)((𝐷 FuncCat 𝐸) UP (𝐶 FuncCat 𝐸))𝑥))𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
30	27, 29	syl 18	1 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  ∅c0 4294  {csn 4594  ⟨cop 4600   × cxp 5660  dom cdm 5662   ↾ cres 5664  Fun wfun 6531  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   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-lan 50270

This theorem is referenced by:  rellan  50286  islan  50288  lanrcl2  50295  lanrcl3  50296