Theorem lanrcl 50242

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 4293	. . . . 5 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅)
3		eqid 2762	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4		eqid 2762	. . . . . 6 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
5		df-ov 7399	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ Lan 𝐸) = ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩)
6	5	eqeq1i 2767	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ ↔ ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅)
7		oveq 7402	. . . . . . . . . 10 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7433	. . . . . . . . . 10 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2813	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Lan 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ∅)
10	6, 9	sylbir 237	. . . . . . . 8 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ∅)
11	10	necon3i 2989	. . . . . . 7 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) ≠ ∅ → ( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅)
12		fvfundmfvn0 6907	. . . . . . . . 9 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Lan ∧ Fun ( Lan ↾ {⟨⟨𝐶, 𝐷⟩, 𝐸⟩})))
13	12	simpld 498	. . . . . . . 8 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Lan )
14		lanfn 50230	. . . . . . . . 9 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6625	. . . . . . . 8 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2872	. . . . . . 7 ⊢ (( Lan ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5689	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → ⟨𝐶, 𝐷⟩ ∈ (V × V))
18		opelxp1 5689	. . . . . . 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 5690	. . . . . . 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 5690	. . . . . . 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 50234	. . . . 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 7413	. . 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 2864	. 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 2762	. . 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 7637	. 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 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  Vcvv 3454  ∅c0 4285  {csn 4582  ⟨cop 4588   × cxp 5645  dom cdm 5647   ↾ cres 5649  Fun wfun 6515  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398   Func cfunc 17887   FuncCat cfuc 17978   UP cup 49794   −∘_F cprcof 49994   Lan clan 50226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-lan 50228

This theorem is referenced by:  rellan  50244  islan  50246  lanrcl2  50253  lanrcl3  50254