Theorem ranval3 50122

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 5413	. . . . . 6 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ⟨𝐷, 𝐸⟩ ∈ V)
6	3, 5	prcofelvv 49871	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V))
7		1st2nd2 7976	. . . 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 50121	. 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 2737	. . . . . . . . 9 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
11	10	fucbas 17925	. . . . . . . 8 ⊢ (𝐶 Func 𝐸) = (Base‘(𝐶 FuncCat 𝐸))
12	2, 11	oppcbas 17679	. . . . . . 7 ⊢ (𝐶 Func 𝐸) = (Base‘𝑃)
13	12	uprcl 49675	. . . . . 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 49675	. . . . . 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 2737	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
20		funcrcl 17825	. . . . . . . . . 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 49877	. . . . . . 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 6839	. . . . . . . 8 ⊢ ( oppFunc ‘𝐾) = ( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))
27		oppfval2 49628	. . . . . . . 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 2784	. . . . . . 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 7377	. . . . 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 2823	. . . 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 840	. . 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 2735	. 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 2775	1 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   × cxp 5624  ‘cfv 6494  (class class class)co 7362  1^st c1st 7935  2^nd c2nd 7936  tpos ctpos 8170  Catccat 17625  oppCatcoppc 17672   Func cfunc 17816   FuncCat cfuc 17907   oppFunc coppf 49613   UP cup 49664   −∘_F cprcof 49864   Ran cran 50097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-tpos 8171  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-map 8770  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-hom 17239  df-cco 17240  df-cat 17629  df-cid 17630  df-oppc 17673  df-func 17820  df-cofu 17822  df-nat 17908  df-fuc 17909  df-xpc 18133  df-curf 18175  df-oppf 49614  df-up 49665  df-swapf 49751  df-fuco 49808  df-prcof 49865  df-ran 50099

This theorem is referenced by:  lmdran  50162