Theorem lanrcl 50943

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 4289	. . . . 5 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅)
3		eqid 2763	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4		eqid 2763	. . . . . 6 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
5		df-ov 7430	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ Lan 𝐸) = ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩)
6	5	eqeq1i 2768	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ ↔ ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅)
7		oveq 7433	. . . . . . . . . 10 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7464	. . . . . . . . . 10 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2814	. . . . . . . . 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 2990	. . . . . . 7 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅)
12		fvfundmfvn0 6933	. . . . . . . . 9 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Lan ∧ Fun ( Lan ↾ {⟨⟨𝐶, 𝐷⟩, 𝐸⟩})))
13	12	simpld 500	. . . . . . . 8 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Lan )
14		lanfn 50931	. . . . . . . . 9 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6651	. . . . . . . 8 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2873	. . . . . . 7 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5696	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → ⟨𝐶, 𝐷⟩ ∈ (V × V))
18		opelxp1 5696	. . . . . . 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 5697	. . . . . . 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 5697	. . . . . . 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 50935	. . . . 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 7444	. . 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 2865	. 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 2763	. . 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 7669	. 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  Vcvv 3453  ∅c0 4280  {csn 4587  ⟨cop 4837   × cxp 5652  dom cdm 5654   ↾ cres 5656  Fun wfun 6540  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429   Func cfunc 18041   FuncCat cfuc 18132   UP cup 50428   −∘_F cprcof 50647   Lan clan 50927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-lan 50929

This theorem is used by:  rellan  50945  islan  50947  lanrcl2  50954  lanrcl3  50955