Theorem ranrcl 50729

Description: Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)

Assertion

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

Proof of Theorem ranrcl

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⟩ Ran 𝐸)𝑋) → 𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋))
2		ne0i 4287	. . . . 5 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅)
3		eqid 2761	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4		eqid 2761	. . . . . 6 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
5		df-ov 7423	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ Ran 𝐸) = ( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩)
6	5	eqeq1i 2766	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Ran 𝐸) = ∅ ↔ ( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅)
7		oveq 7426	. . . . . . . . . 10 ⊢ ((⟨𝐶, 𝐷⟩ Ran 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7457	. . . . . . . . . 10 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2812	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Ran 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = ∅)
10	6, 9	sylbir 238	. . . . . . . 8 ⊢ (( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = ∅)
11	10	necon3i 2988	. . . . . . 7 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → ( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅)
12		fvfundmfvn0 6925	. . . . . . . . 9 ⊢ (( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Ran ∧ Fun ( Ran ↾ {⟨⟨𝐶, 𝐷⟩, 𝐸⟩})))
13	12	simpld 500	. . . . . . . 8 ⊢ (( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Ran )
14		ranfn 50717	. . . . . . . . 9 ⊢ Ran Fn ((V × V) × V)
15	14	fndmi 6643	. . . . . . . 8 ⊢ dom Ran = ((V × V) × V)
16	13, 15	eleqtrdi 2871	. . . . . . 7 ⊢ (( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5693	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → ⟨𝐶, 𝐷⟩ ∈ (V × V))
18		opelxp1 5693	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐶 ∈ V)
19	11, 16, 17, 18	4syl 20	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → 𝐶 ∈ V)
20		opelxp2 5694	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐷 ∈ V)
21	11, 16, 17, 20	4syl 20	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → 𝐷 ∈ V)
22		opelxp2 5694	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
23	11, 16, 22	3syl 19	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → 𝐸 ∈ V)
24		eqid 2761	. . . . . 6 ⊢ (oppCat‘(𝐷 FuncCat 𝐸)) = (oppCat‘(𝐷 FuncCat 𝐸))
25		eqid 2761	. . . . . 6 ⊢ (oppCat‘(𝐶 FuncCat 𝐸)) = (oppCat‘(𝐶 FuncCat 𝐸))
26	3, 4, 19, 21, 23, 24, 25	ranfval 50721	. . . . 5 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → (⟨𝐶, 𝐷⟩ Ran 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘(𝐶 FuncCat 𝐸)))𝑥)))
27	2, 26	syl 18	. . . 4 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (⟨𝐶, 𝐷⟩ Ran 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘(𝐶 FuncCat 𝐸)))𝑥)))
28	27	oveqd 7437	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (𝐹(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘(𝐶 FuncCat 𝐸)))𝑥))𝑋))
29	1, 28	eleqtrd 2863	. 2 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → 𝐿 ∈ (𝐹(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘(𝐶 FuncCat 𝐸)))𝑥))𝑋))
30		eqid 2761	. . 3 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘(𝐶 FuncCat 𝐸)))𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘(𝐶 FuncCat 𝐸)))𝑥))
31	30	elmpocl 7662	. 2 ⊢ (𝐿 ∈ (𝐹(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘(𝐶 FuncCat 𝐸)))𝑥))𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
32	29, 31	syl 18	1 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  Vcvv 3451  ∅c0 4279  {csn 4584  ⟨cop 4590   × cxp 5649  dom cdm 5651   ↾ cres 5653  Fun wfun 6532  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  oppCatcoppc 17885   Func cfunc 18029   FuncCat cfuc 18120   oppFunc coppf 50229   UP cup 50280   −∘_F cprcof 50480   Ran cran 50713

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-ran 50715

This theorem is used by:  relran  50731  isran  50735  ranrcl2  50743  ranrcl3  50744