Theorem ranrcl 49358

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

Assertion

Ref	Expression
ranrcl	⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)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⟩Ran𝐸)𝑋) → 𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋))
2		ne0i 4314	. . . . 5 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) ≠ ∅)
3		eqid 2734	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4		eqid 2734	. . . . . 6 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
5		df-ov 7403	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩Ran𝐸) = (Ran‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩)
6	5	eqeq1i 2739	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩Ran𝐸) = ∅ ↔ (Ran‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅)
7		oveq 7406	. . . . . . . . . 10 ⊢ ((⟨𝐶, 𝐷⟩Ran𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7437	. . . . . . . . . 10 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2785	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩Ran𝐸) = ∅ → (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) = ∅)
10	6, 9	sylbir 235	. . . . . . . 8 ⊢ ((Ran‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) = ∅ → (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) = ∅)
11	10	necon3i 2963	. . . . . . 7 ⊢ ((𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) ≠ ∅ → (Ran‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅)
12		fvfundmfvn0 6916	. . . . . . . . 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 49348	. . . . . . . . 9 ⊢ Ran Fn ((V × V) × V)
15	14	fndmi 6639	. . . . . . . 8 ⊢ dom Ran = ((V × V) × V)
16	13, 15	eleqtrdi 2843	. . . . . . 7 ⊢ ((Ran‘⟨⟨𝐶, 𝐷⟩, 𝐸⟩) ≠ ∅ → ⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5694	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → ⟨𝐶, 𝐷⟩ ∈ (V × V))
18		opelxp1 5694	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐶 ∈ V)
19	11, 16, 17, 18	4syl 19	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) ≠ ∅ → 𝐶 ∈ V)
20		opelxp2 5695	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ ∈ (V × V) → 𝐷 ∈ V)
21	11, 16, 17, 20	4syl 19	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) ≠ ∅ → 𝐷 ∈ V)
22		opelxp2 5695	. . . . . . 7 ⊢ (⟨⟨𝐶, 𝐷⟩, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
23	11, 16, 22	3syl 18	. . . . . 6 ⊢ ((𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) ≠ ∅ → 𝐸 ∈ V)
24		eqid 2734	. . . . . 6 ⊢ (oppCat‘(𝐷 FuncCat 𝐸)) = (oppCat‘(𝐷 FuncCat 𝐸))
25		eqid 2734	. . . . . 6 ⊢ (oppCat‘(𝐶 FuncCat 𝐸)) = (oppCat‘(𝐶 FuncCat 𝐸))
26	3, 4, 19, 21, 23, 24, 25	ranfval 49352	. . . . 5 ⊢ ((𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) ≠ ∅ → (⟨𝐶, 𝐷⟩Ran𝐸) = (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⟩Ran𝐸)𝑋) → (⟨𝐶, 𝐷⟩Ran𝐸) = (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 7417	. . 3 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)𝑋) → (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)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 2835	. 2 ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩Ran𝐸)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 2734	. . 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 7643	. 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⟩Ran𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  Vcvv 3457  ∅c0 4306  {csn 4599  ⟨cop 4605   × cxp 5650  dom cdm 5652   ↾ cres 5654  Fun wfun 6522  ‘cfv 6528  (class class class)co 7400   ∈ cmpo 7402  oppCatcoppc 17710   Func cfunc 17854   FuncCat cfuc 17945  oppFunccoppf 48950  UPcup 48974   −∘_F cprcof 49147  Rancran 49344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-ran 49346

This theorem is referenced by:  relran  49360  isran  49364  ranrcl2  49371  ranrcl3  49372