Theorem ranrcl 51136

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 4290	. . . . 5 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Ran 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 7454	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ Ran 𝐸) = ( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩)
6	5	eqeq1i 2768	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Ran 𝐸) = ∅ ↔ ( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅)
7		oveq 7457	. . . . . . . . . 10 ⊢ ((⟨𝐶, 𝐷⟩ Ran 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7488	. . . . . . . . . 10 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2814	. . . . . . . . 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 2991	. . . . . . 7 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → ( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅)
12		fvfundmfvn0 6956	. . . . . . . . 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 51124	. . . . . . . . 9 ⊢ Ran Fn ((V × V) × V)
15	14	fndmi 6673	. . . . . . . 8 ⊢ dom Ran = ((V × V) × V)
16	13, 15	eleqtrdi 2874	. . . . . . 7 ⊢ (( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5700	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → ⟨𝐶, 𝐷⟩ ∈ (V × V))
18		opelxp1 5700	. . . . . . 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 5701	. . . . . . 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 5701	. . . . . . 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 2763	. . . . . 6 ⊢ (oppCat‘(𝐷 FuncCat 𝐸)) = (oppCat‘(𝐷 FuncCat 𝐸))
25		eqid 2763	. . . . . 6 ⊢ (oppCat‘(𝐶 FuncCat 𝐸)) = (oppCat‘(𝐶 FuncCat 𝐸))
26	3, 4, 19, 21, 23, 24, 25	ranfval 51128	. . . . 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 7468	. . 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 2866	. 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 2763	. . 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 7694	. 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 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  {csn 4588  ⟨cop 4840   × cxp 5656  dom cdm 5658   ↾ cres 5660  Fun wfun 6560  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  oppCatcoppc 17935   Func cfunc 18079   FuncCat cfuc 18170   oppFunc coppf 50569   UP cup 50620   −∘_F cprcof 50839   Ran cran 51120

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-ran 51122

This theorem is used by:  relran  51138  isran  51142  ranrcl2  51150  ranrcl3  51151