Theorem lanrcl 49732

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 4288	. . . . 5 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅)
3		eqid 2731	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4		eqid 2731	. . . . . 6 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
5		df-ov 7349	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ Lan 𝐸) = ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩)
6	5	eqeq1i 2736	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ ↔ ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅)
7		oveq 7352	. . . . . . . . . 10 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7383	. . . . . . . . . 10 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2782	. . . . . . . . 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 2960	. . . . . . 7 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅)
12		fvfundmfvn0 6862	. . . . . . . . 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 49720	. . . . . . . . 9 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6585	. . . . . . . 8 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2841	. . . . . . 7 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5656	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → ⟨𝐶, 𝐷⟩ ∈ (V × V))
18		opelxp1 5656	. . . . . . 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 5657	. . . . . . 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 5657	. . . . . . 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 49724	. . . . 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 7363	. . 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 2833	. 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 2731	. . 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 7587	. 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 1541   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436  ∅c0 4280  {csn 4573  ⟨cop 4579   × cxp 5612  dom cdm 5614   ↾ cres 5616  Fun wfun 6475  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348   Func cfunc 17761   FuncCat cfuc 17852   UP cup 49284   −∘_F cprcof 49484   Lan clan 49716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-lan 49718

This theorem is referenced by:  rellan  49734  islan  49736  lanrcl2  49743  lanrcl3  49744