Theorem relran 49879

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 5748	. . 3 ⊢ Rel ∅
2		releq 5726	. . 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 4305	. . 3 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋))
5		relup 49438	. . . . 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 7364	. . . . . . . . . . . 12 ⊢ ((𝑃 Ran 𝐸) = ∅ → (𝐹(𝑃 Ran 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7395	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2787	. . . . . . . . . . 11 ⊢ ((𝑃 Ran 𝐸) = ∅ → (𝐹(𝑃 Ran 𝐸)𝑋) = ∅)
10	9	necon3i 2964	. . . . . . . . . 10 ⊢ ((𝐹(𝑃 Ran 𝐸)𝑋) ≠ ∅ → (𝑃 Ran 𝐸) ≠ ∅)
11		n0 4305	. . . . . . . . . . 11 ⊢ ((𝑃 Ran 𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃 Ran 𝐸))
12		df-ran 49863	. . . . . . . . . . . . . 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 7600	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑃 Ran 𝐸) → 𝑃 ∈ (V × V))
14		1st2nd2 7972	. . . . . . . . . . . . 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 1931	. . . . . . . . . . 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 7373	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝑃 Ran 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸))
20	19	oveqd 7375	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (𝐹(𝑃 Ran 𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋))
21		eqid 2736	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2736	. . . . . . . 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 2838	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸)𝑋))
25		ranrcl 49877	. . . . . . . . . 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 5412	. . . . . . . . . . 11 ⊢ ⟨(2nd ‘𝑃), 𝐸⟩ ∈ V
30	29	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → ⟨(2nd ‘𝑃), 𝐸⟩ ∈ V)
31	27, 30	prcofelvv 49635	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃 Ran 𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) ∈ (V × V))
32		1st2nd2 7972	. . . . . . . . 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 2736	. . . . . . . 8 ⊢ (oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) = (oppCat‘((2nd ‘𝑃) FuncCat 𝐸))
35		eqid 2736	. . . . . . . 8 ⊢ (oppCat‘((1st ‘𝑃) FuncCat 𝐸)) = (oppCat‘((1st ‘𝑃) FuncCat 𝐸))
36	21, 22, 27, 28, 33, 34, 35	ranval 49875	. . . . . . 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 2771	. . . . . 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 5728	. . . . 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 1931	. . 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 3015	1 ⊢ Rel (𝐹(𝑃 Ran 𝐸)𝑋)

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440  ⦋csb 3849  ∅c0 4285  ⟨cop 4586   × cxp 5622  Rel wrel 5629  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932  tpos ctpos 8167  oppCatcoppc 17634   Func cfunc 17778   FuncCat cfuc 17869   oppFunc coppf 49377   UP cup 49428   −∘_F cprcof 49628   Ran cran 49861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-hom 17201  df-cco 17202  df-cat 17591  df-cid 17592  df-func 17782  df-cofu 17784  df-nat 17870  df-fuc 17871  df-xpc 18095  df-curf 18137  df-oppf 49378  df-up 49429  df-swapf 49515  df-fuco 49572  df-prcof 49629  df-ran 49863

This theorem is referenced by: (None)