Theorem ranval3 49610

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 5426	. . . . . 6 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ⟨𝐷, 𝐸⟩ ∈ V)
6	3, 5	prcofelvv 49359	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V))
7		1st2nd2 8009	. . . 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 49609	. 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 2730	. . . . . . . . 9 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
11	10	fucbas 17931	. . . . . . . 8 ⊢ (𝐶 Func 𝐸) = (Base‘(𝐶 FuncCat 𝐸))
12	2, 11	oppcbas 17685	. . . . . . 7 ⊢ (𝐶 Func 𝐸) = (Base‘𝑃)
13	12	uprcl 49163	. . . . . 6 ⊢ (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) → (( oppFunc ‘𝐾) ∈ (𝑂 Func 𝑃) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
14	13	simprd 495	. . . . 5 ⊢ (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
15	14	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋)) → 𝑋 ∈ (𝐶 Func 𝐸))
16	12	uprcl 49163	. . . . . 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 495	. . . . 5 ⊢ (𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
18	17	adantl 481	. . . 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 2730	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
20		funcrcl 17831	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
21	20	simprd 495	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝐶 Func 𝐸) → 𝐸 ∈ Cat)
22	21	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → 𝐸 ∈ Cat)
23		simpl 482	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → 𝐹 ∈ (𝐶 Func 𝐷))
24	19, 22, 10, 23	prcoffunca 49365	. . . . . . 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 6863	. . . . . . . 8 ⊢ ( oppFunc ‘𝐾) = ( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))
27		oppfval2 49116	. . . . . . . 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 2777	. . . . . . 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 7404	. . . . 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 2815	. . . 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 839	. . 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 2728	. 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 2768	1 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4597   × cxp 5638  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969  tpos ctpos 8206  Catccat 17631  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-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-oppc 17679  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:  lmdran  49650