Theorem relran 49603

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 5764	. . 3 ⊢ Rel ∅
2		releq 5741	. . 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 4318	. . 3 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋))
5		relup 49162	. . . . 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 4306	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅)
7		oveq 7395	. . . . . . . . . . . 12 ⊢ ((𝑃 Ran 𝐸) = ∅ → (𝐹(𝑃 Ran 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7426	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2781	. . . . . . . . . . 11 ⊢ ((𝑃 Ran 𝐸) = ∅ → (𝐹(𝑃 Ran 𝐸)𝑋) = ∅)
10	9	necon3i 2958	. . . . . . . . . 10 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ → (𝑃 Ran 𝐸) ≠ ∅)
11		n0 4318	. . . . . . . . . . 11 ⊢ ((𝑃 Ran 𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃 Ran 𝐸))
12		df-ran 49587	. . . . . . . . . . . . . 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 7633	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑃 Ran 𝐸) → 𝑃 ∈ (V × V))
14		1st2nd2 8009	. . . . . . . . . . . . 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 7404	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝑃 Ran 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸))
20	19	oveqd 7406	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹(𝑃 Ran 𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋))
21		eqid 2730	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2730	. . . . . . . 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 2831	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋))
25		ranrcl 49601	. . . . . . . . . 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 5426	. . . . . . . . . . 11 ⊢ ⟨(2nd ‘𝑃), 𝐸⟩ ∈ V
30	29	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → ⟨(2nd ‘𝑃), 𝐸⟩ ∈ V)
31	27, 30	prcofelvv 49359	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) ∈ (V × V))
32		1st2nd2 8009	. . . . . . . . 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 2730	. . . . . . . 8 ⊢ (oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) = (oppCat‘((2nd ‘𝑃) FuncCat 𝐸))
35		eqid 2730	. . . . . . . 8 ⊢ (oppCat‘((1st ‘𝑃) FuncCat 𝐸)) = (oppCat‘((1st ‘𝑃) FuncCat 𝐸))
36	21, 22, 27, 28, 33, 34, 35	ranval 49599	. . . . . . 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 2765	. . . . . 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 5743	. . . . 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 3009	1 ⊢ Rel (𝐹(𝑃 Ran 𝐸)𝑋)

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450  ⦋csb 3864  ∅c0 4298  ⟨cop 4597   × cxp 5638  Rel wrel 5645  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  1^st c1st 7968  2^nd c2nd 7969  tpos ctpos 8206  oppCatcoppc 17678   Func cfunc 17822   FuncCat cfuc 17913   oppFunc coppf 49101   UP cup 49152   −∘_F cprcof 49352   Ran cran 49585

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-map 8803  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-uz 12800  df-fz 13475  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-func 17826  df-cofu 17828  df-nat 17914  df-fuc 17915  df-xpc 18139  df-curf 18181  df-oppf 49102  df-up 49153  df-swapf 49239  df-fuco 49296  df-prcof 49353  df-ran 49587

This theorem is referenced by: (None)