Theorem lanrcl 49600

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 22	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋))
2		ne0i 4306	. . . . 5 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅)
3		eqid 2730	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4		eqid 2730	. . . . . 6 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
5		df-ov 7392	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ Lan 𝐸) = ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩)
6	5	eqeq1i 2735	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ ↔ ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅)
7		oveq 7395	. . . . . . . . . 10 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7426	. . . . . . . . . 10 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2781	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ∅)
10	6, 9	sylbir 235	. . . . . . . 8 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ∅)
11	10	necon3i 2958	. . . . . . 7 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅)
12		fvfundmfvn0 6903	. . . . . . . . 9 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Lan ∧ Fun ( Lan ↾ {⟨⟨𝐶, 𝐷⟩, 𝐸⟩})))
13	12	simpld 494	. . . . . . . 8 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Lan )
14		lanfn 49588	. . . . . . . . 9 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6624	. . . . . . . 8 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2839	. . . . . . 7 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5682	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → ⟨𝐶, 𝐷⟩ ∈ (V × V))
18		opelxp1 5682	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐶 ∈ V)
19	11, 16, 17, 18	4syl 19	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → 𝐶 ∈ V)
20		opelxp2 5683	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐷 ∈ V)
21	11, 16, 17, 20	4syl 19	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → 𝐷 ∈ V)
22		opelxp2 5683	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
23	11, 16, 22	3syl 18	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → 𝐸 ∈ V)
24	3, 4, 19, 21, 23	lanfval 49592	. . . . 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 17	. . . 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 7406	. . 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 2831	. 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 2730	. . 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 7632	. 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 17	1 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450  ∅c0 4298  {csn 4591  ⟨cop 4597   × cxp 5638  dom cdm 5640   ↾ cres 5642  Fun wfun 6507  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391   Func cfunc 17822   FuncCat cfuc 17913   UP cup 49152   −∘_F cprcof 49352   Lan clan 49584

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-lan 49586

This theorem is referenced by:  rellan  49602  islan  49604  lanrcl2  49611  lanrcl3  49612