Theorem relran 50245

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 5771	. . 3 ⊢ Rel ∅
2		releq 5749	. . 3 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) = ∅ → (Rel (𝐹(𝑃 Ran 𝐸)𝑋) ↔ Rel ∅))
3	1, 2	mpbiri 260	. 2 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) = ∅ → Rel (𝐹(𝑃 Ran 𝐸)𝑋))
4		n0 4305	. . 3 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋))
5		relup 49804	. . . . 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 4293	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅)
7		oveq 7402	. . . . . . . . . . . 12 ⊢ ((𝑃 Ran 𝐸) = ∅ → (𝐹(𝑃 Ran 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7433	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2813	. . . . . . . . . . 11 ⊢ ((𝑃 Ran 𝐸) = ∅ → (𝐹(𝑃 Ran 𝐸)𝑋) = ∅)
10	9	necon3i 2989	. . . . . . . . . 10 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ → (𝑃 Ran 𝐸) ≠ ∅)
11		n0 4305	. . . . . . . . . . 11 ⊢ ((𝑃 Ran 𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃 Ran 𝐸))
12		df-ran 50229	. . . . . . . . . . . . . 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 7638	. . . . . . . . . . . . 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 1950	. . . . . . . . . . 11 ⊢ (∃𝑥 𝑥 ∈ (𝑃 Ran 𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
17	11, 16	sylbi 219	. . . . . . . . . 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 7411	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝑃 Ran 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸))
20	19	oveqd 7413	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹(𝑃 Ran 𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋))
21		eqid 2762	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2762	. . . . . . . 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 2864	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋))
25		ranrcl 50243	. . . . . . . . . 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 498	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)))
28	26	simprd 499	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝑋 ∈ ((1st ‘𝑃) Func 𝐸))
29		opex 5431	. . . . . . . . . . 11 ⊢ ⟨(2nd ‘𝑃), 𝐸⟩ ∈ V
30	29	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → ⟨(2nd ‘𝑃), 𝐸⟩ ∈ V)
31	27, 30	prcofelvv 50001	. . . . . . . . 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 2762	. . . . . . . 8 ⊢ (oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) = (oppCat‘((2nd ‘𝑃) FuncCat 𝐸))
35		eqid 2762	. . . . . . . 8 ⊢ (oppCat‘((1st ‘𝑃) FuncCat 𝐸)) = (oppCat‘((1st ‘𝑃) FuncCat 𝐸))
36	21, 22, 27, 28, 33, 34, 35	ranval 50241	. . . . . . 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 2797	. . . . . 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 5751	. . . . 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 260	. . . 4 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → Rel (𝐹(𝑃 Ran 𝐸)𝑋))
40	39	exlimiv 1950	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → Rel (𝐹(𝑃 Ran 𝐸)𝑋))
41	4, 40	sylbi 219	. 2 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ → Rel (𝐹(𝑃 Ran 𝐸)𝑋))
42	3, 41	pm2.61ine 3040	1 ⊢ Rel (𝐹(𝑃 Ran 𝐸)𝑋)

Syntax hints:   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142   ≠ wne 2957  Vcvv 3454  ⦋csb 3852  ∅c0 4285  ⟨cop 4588   × cxp 5645  Rel wrel 5652  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  1^st c1st 7968  2^nd c2nd 7969  tpos ctpos 8205  oppCatcoppc 17743   Func cfunc 17887   FuncCat cfuc 17978   oppFunc coppf 49743   UP cup 49794   −∘_F cprcof 49994   Ran cran 50227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-tpos 8206  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-map 8810  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-fz 13513  df-struct 17183  df-slot 17218  df-ndx 17230  df-base 17246  df-hom 17310  df-cco 17311  df-cat 17700  df-cid 17701  df-func 17891  df-cofu 17893  df-nat 17979  df-fuc 17980  df-xpc 18204  df-curf 18246  df-oppf 49744  df-up 49795  df-swapf 49881  df-fuco 49938  df-prcof 49995  df-ran 50229

This theorem is referenced by: (None)