Theorem relran 49613

Description: The set of right Kan extensions is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)

Assertion

Ref	Expression
relran	⊢ Rel (𝐹(𝑃 Ran 𝐸)𝑋)

Proof of Theorem relran

Dummy variables 𝑓 𝑥 𝑐 𝑑 𝑒 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rel0 5742	. . 3 ⊢ Rel ∅
2		releq 5720	. . 3 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) = ∅ → (Rel (𝐹(𝑃 Ran 𝐸)𝑋) ↔ Rel ∅))
3	1, 2	mpbiri 258	. 2 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) = ∅ → Rel (𝐹(𝑃 Ran 𝐸)𝑋))
4		n0 4304	. . 3 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋))
5		relup 49172	. . . . 5 ⊢ Rel (⟨(1st ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))⟩((oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) UP (oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑋)
6		ne0i 4292	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅)
7		oveq 7355	. . . . . . . . . . . 12 ⊢ ((𝑃 Ran 𝐸) = ∅ → (𝐹(𝑃 Ran 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7386	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2780	. . . . . . . . . . 11 ⊢ ((𝑃 Ran 𝐸) = ∅ → (𝐹(𝑃 Ran 𝐸)𝑋) = ∅)
10	9	necon3i 2957	. . . . . . . . . 10 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ → (𝑃 Ran 𝐸) ≠ ∅)
11		n0 4304	. . . . . . . . . . 11 ⊢ ((𝑃 Ran 𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃 Ran 𝐸))
12		df-ran 49597	. . . . . . . . . . . . . 14 ⊢ Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))
13	12	elmpocl1 7591	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑃 Ran 𝐸) → 𝑃 ∈ (V × V))
14		1st2nd2 7963	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (V × V) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
15	13, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑃 Ran 𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
16	15	exlimiv 1930	. . . . . . . . . . 11 ⊢ (∃𝑥 𝑥 ∈ (𝑃 Ran 𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
17	11, 16	sylbi 217	. . . . . . . . . 10 ⊢ ((𝑃 Ran 𝐸) ≠ ∅ → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
18	6, 10, 17	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
19	18	oveq1d 7364	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝑃 Ran 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸))
20	19	oveqd 7366	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹(𝑃 Ran 𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋))
21		eqid 2729	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2729	. . . . . . . 8 ⊢ ((1st ‘𝑃) FuncCat 𝐸) = ((1st ‘𝑃) FuncCat 𝐸)
23		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋))
24	23, 20	eleqtrd 2830	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋))
25		ranrcl 49611	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋) → (𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∧ 𝑋 ∈ ((1st ‘𝑃) Func 𝐸)))
26	24, 25	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∧ 𝑋 ∈ ((1st ‘𝑃) Func 𝐸)))
27	26	simpld 494	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)))
28	26	simprd 495	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝑋 ∈ ((1st ‘𝑃) Func 𝐸))
29		opex 5407	. . . . . . . . . . 11 ⊢ ⟨(2nd ‘𝑃), 𝐸⟩ ∈ V
30	29	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → ⟨(2nd ‘𝑃), 𝐸⟩ ∈ V)
31	27, 30	prcofelvv 49369	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) ∈ (V × V))
32		1st2nd2 7963	. . . . . . . . 9 ⊢ ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) ∈ (V × V) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))⟩)
33	31, 32	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))⟩)
34		eqid 2729	. . . . . . . 8 ⊢ (oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) = (oppCat‘((2nd ‘𝑃) FuncCat 𝐸))
35		eqid 2729	. . . . . . . 8 ⊢ (oppCat‘((1st ‘𝑃) FuncCat 𝐸)) = (oppCat‘((1st ‘𝑃) FuncCat 𝐸))
36	21, 22, 27, 28, 33, 34, 35	ranval 49609	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋) = (⟨(1st ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))⟩((oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) UP (oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑋))
37	20, 36	eqtrd 2764	. . . . . 6 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹(𝑃 Ran 𝐸)𝑋) = (⟨(1st ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))⟩((oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) UP (oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑋))
38	37	releqd 5722	. . . . 5 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (Rel (𝐹(𝑃 Ran 𝐸)𝑋) ↔ Rel (⟨(1st ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))⟩((oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) UP (oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑋)))
39	5, 38	mpbiri 258	. . . 4 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → Rel (𝐹(𝑃 Ran 𝐸)𝑋))
40	39	exlimiv 1930	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → Rel (𝐹(𝑃 Ran 𝐸)𝑋))
41	4, 40	sylbi 217	. 2 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ → Rel (𝐹(𝑃 Ran 𝐸)𝑋))
42	3, 41	pm2.61ine 3008	1 ⊢ Rel (𝐹(𝑃 Ran 𝐸)𝑋)

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436  ⦋csb 3851  ∅c0 4284  ⟨cop 4583   × cxp 5617  Rel wrel 5624  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  1^st c1st 7922  2^nd c2nd 7923  tpos ctpos 8158  oppCatcoppc 17617   Func cfunc 17761   FuncCat cfuc 17852   oppFunc coppf 49111   UP cup 49162   −∘_F cprcof 49362   Ran cran 49595

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-tpos 8159  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-cat 17574  df-cid 17575  df-func 17765  df-cofu 17767  df-nat 17853  df-fuc 17854  df-xpc 18078  df-curf 18120  df-oppf 49112  df-up 49163  df-swapf 49249  df-fuco 49306  df-prcof 49363  df-ran 49597

This theorem is referenced by: (None)