Theorem ranval3 49897

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 5412	. . . . . 6 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ⟨𝐷, 𝐸⟩ ∈ V)
6	3, 5	prcofelvv 49646	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V))
7		1st2nd2 7972	. . . 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 49896	. 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 2736	. . . . . . . . 9 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
11	10	fucbas 17889	. . . . . . . 8 ⊢ (𝐶 Func 𝐸) = (Base‘(𝐶 FuncCat 𝐸))
12	2, 11	oppcbas 17643	. . . . . . 7 ⊢ (𝐶 Func 𝐸) = (Base‘𝑃)
13	12	uprcl 49450	. . . . . 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 49450	. . . . . 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 2736	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
20		funcrcl 17789	. . . . . . . . . 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 49652	. . . . . . 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 6837	. . . . . . . 8 ⊢ ( oppFunc ‘𝐾) = ( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))
27		oppfval2 49403	. . . . . . . 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 2783	. . . . . . 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 7373	. . . . 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 2822	. . . 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 2734	. 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 2774	1 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ⟨cop 4586   × cxp 5622  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  tpos ctpos 8167  Catccat 17589  oppCatcoppc 17636   Func cfunc 17780   FuncCat cfuc 17871   oppFunc coppf 49388   UP cup 49439   −∘_F cprcof 49639   Ran cran 49872

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 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 8767  df-ixp 8838  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-uz 12754  df-fz 13426  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-hom 17203  df-cco 17204  df-cat 17593  df-cid 17594  df-oppc 17637  df-func 17784  df-cofu 17786  df-nat 17872  df-fuc 17873  df-xpc 18097  df-curf 18139  df-oppf 49389  df-up 49440  df-swapf 49526  df-fuco 49583  df-prcof 49640  df-ran 49874

This theorem is referenced by:  lmdran  49937