Theorem ranrcl 49604

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 22	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → 𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋))
2		ne0i 4300	. . . . 5 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅)
3		eqid 2729	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4		eqid 2729	. . . . . 6 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
5		df-ov 7372	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ Ran 𝐸) = ( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩)
6	5	eqeq1i 2734	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Ran 𝐸) = ∅ ↔ ( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅)
7		oveq 7375	. . . . . . . . . 10 ⊢ ((⟨𝐶, 𝐷⟩ Ran 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7406	. . . . . . . . . 10 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2780	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ Ran 𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = ∅)
10	6, 9	sylbir 235	. . . . . . . 8 ⊢ (( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅ → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = ∅)
11	10	necon3i 2957	. . . . . . 7 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → ( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅)
12		fvfundmfvn0 6883	. . . . . . . . 9 ⊢ (( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Ran ∧ Fun ( Ran ↾ {⟨⟨𝐶, 𝐷⟩, 𝐸⟩})))
13	12	simpld 494	. . . . . . . 8 ⊢ (( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ dom Ran )
14		ranfn 49592	. . . . . . . . 9 ⊢ Ran Fn ((V × V) × V)
15	14	fndmi 6604	. . . . . . . 8 ⊢ dom Ran = ((V × V) × V)
16	13, 15	eleqtrdi 2838	. . . . . . 7 ⊢ (( Ran ‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5673	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → ⟨𝐶, 𝐷⟩ ∈ (V × V))
18		opelxp1 5673	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐶 ∈ V)
19	11, 16, 17, 18	4syl 19	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → 𝐶 ∈ V)
20		opelxp2 5674	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐷 ∈ V)
21	11, 16, 17, 20	4syl 19	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → 𝐷 ∈ V)
22		opelxp2 5674	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
23	11, 16, 22	3syl 18	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) ≠ ∅ → 𝐸 ∈ V)
24		eqid 2729	. . . . . 6 ⊢ (oppCat‘(𝐷 FuncCat 𝐸)) = (oppCat‘(𝐷 FuncCat 𝐸))
25		eqid 2729	. . . . . 6 ⊢ (oppCat‘(𝐶 FuncCat 𝐸)) = (oppCat‘(𝐶 FuncCat 𝐸))
26	3, 4, 19, 21, 23, 24, 25	ranfval 49596	. . . . 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 17	. . . 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 7386	. . 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 2830	. 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 2729	. . 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 7610	. 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 17	1 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3444  ∅c0 4292  {csn 4585  ⟨cop 4591   × cxp 5629  dom cdm 5631   ↾ cres 5633  Fun wfun 6493  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  oppCatcoppc 17652   Func cfunc 17796   FuncCat cfuc 17887   oppFunc coppf 49104   UP cup 49155   −∘_F cprcof 49355   Ran cran 49588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-ran 49590

This theorem is referenced by:  relran  49606  isran  49610  ranrcl2  49618  ranrcl3  49619