Theorem ranval3 50129

Description: The set of right Kan extensions is the set of universal pairs. (Contributed by Zhi Wang, 26-Nov-2025.)

Hypotheses

Ref	Expression
ranval3.o	⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))
ranval3.p	⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))
ranval3.k	⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)

Assertion

Ref	Expression
ranval3	⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))

Proof of Theorem ranval3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ranval3.o	. . 3 ⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))
2		ranval3.p	. . 3 ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))
3		id 22	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 ∈ (𝐶 Func 𝐷))
4		opex 5404	. . . . . 6 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ⟨𝐷, 𝐸⟩ ∈ V)
6	3, 5	prcofelvv 49878	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V))
7		1st2nd2 7971	. . . 4 ⊢ ((⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
8	6, 7	syl 17	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
9	1, 2, 8, 3	ranval2 50128	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋))
10		eqid 2739	. . . . . . . . 9 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
11	10	fucbas 17922	. . . . . . . 8 ⊢ (𝐶 Func 𝐸) = (Base‘(𝐶 FuncCat 𝐸))
12	2, 11	oppcbas 17676	. . . . . . 7 ⊢ (𝐶 Func 𝐸) = (Base‘𝑃)
13	12	uprcl 49682	. . . . . 6 ⊢ (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) → (( oppFunc ‘𝐾) ∈ (𝑂 Func 𝑃) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
14	13	simprd 496	. . . . 5 ⊢ (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
15	14	adantl 482	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋)) → 𝑋 ∈ (𝐶 Func 𝐸))
16	12	uprcl 49682	. . . . . 6 ⊢ (𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋) → (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩ ∈ (𝑂 Func 𝑃) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
17	16	simprd 496	. . . . 5 ⊢ (𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
18	17	adantl 482	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋)) → 𝑋 ∈ (𝐶 Func 𝐸))
19		eqid 2739	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
20		funcrcl 17822	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
21	20	simprd 496	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝐶 Func 𝐸) → 𝐸 ∈ Cat)
22	21	adantl 482	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → 𝐸 ∈ Cat)
23		simpl 483	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → 𝐹 ∈ (𝐶 Func 𝐷))
24	19, 22, 10, 23	prcoffunca 49884	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func (𝐶 FuncCat 𝐸)))
25		ranval3.k	. . . . . . . . 9 ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)
26	25	fveq2i 6831	. . . . . . . 8 ⊢ ( oppFunc ‘𝐾) = ( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))
27		oppfval2 49635	. . . . . . . 8 ⊢ ((⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func (𝐶 FuncCat 𝐸)) → ( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
28	26, 27	eqtrid 2786	. . . . . . 7 ⊢ ((⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func (𝐶 FuncCat 𝐸)) → ( oppFunc ‘𝐾) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
29	24, 28	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → ( oppFunc ‘𝐾) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
30	29	oveq1d 7372	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) = (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋))
31	30	eleq2d 2825	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) ↔ 𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋)))
32	15, 18, 31	bibiad 845	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) ↔ 𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋)))
33	32	eqrdv 2737	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) = (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋))
34	9, 33	eqtr4d 2777	1 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4562   × cxp 5617  ‘cfv 6486  (class class class)co 7357  1^st c1st 7930  2^nd c2nd 7931  tpos ctpos 8166  Catccat 17622  oppCatcoppc 17669   Func cfunc 17813   FuncCat cfuc 17904   oppFunc coppf 49620   UP cup 49671   −∘_F cprcof 49871   Ran cran 50104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  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 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-uz 12781  df-fz 13454  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-hom 17236  df-cco 17237  df-cat 17626  df-cid 17627  df-oppc 17670  df-func 17817  df-cofu 17819  df-nat 17905  df-fuc 17906  df-xpc 18130  df-curf 18172  df-oppf 49621  df-up 49672  df-swapf 49758  df-fuco 49815  df-prcof 49872  df-ran 50106

This theorem is referenced by:  lmdran  50169