Theorem ranfval 49352

Description: Value of the function generating the set of right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)

Hypotheses

Ref	Expression
lanfval.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
lanfval.s	⊢ 𝑆 = (𝐶 FuncCat 𝐸)
lanfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
lanfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
lanfval.e	⊢ (𝜑 → 𝐸 ∈ 𝑊)
ranfval.o	⊢ 𝑂 = (oppCat‘𝑅)
ranfval.p	⊢ 𝑃 = (oppCat‘𝑆)

Assertion

Ref	Expression
ranfval	⊢ (𝜑 → (⟨𝐶, 𝐷⟩Ran𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((oppFunc‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂UP𝑃)𝑥)))

Distinct variable groups:   𝐶,𝑓,𝑥   𝐷,𝑓,𝑥   𝑓,𝐸,𝑥   𝜑,𝑓,𝑥

Allowed substitution hints:   𝑃(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝑆(𝑥,𝑓)   𝑈(𝑥,𝑓)   𝑂(𝑥,𝑓)   𝑉(𝑥,𝑓)   𝑊(𝑥,𝑓)

Proof of Theorem ranfval

Dummy variables 𝑐 𝑑 𝑒 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ran 49346	. . 3 ⊢ 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)))
2	1	a1i 11	. 2 ⊢ (𝜑 → 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))))
3		fvexd 6888	. . 3 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → (1st ‘𝑝) ∈ V)
4		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → 𝑝 = ⟨𝐶, 𝐷⟩)
5	4	fveq2d 6877	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → (1st ‘𝑝) = (1st ‘⟨𝐶, 𝐷⟩))
6		lanfval.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
7		lanfval.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
8		op1stg 7995	. . . . . 6 ⊢ ((𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
9	6, 7, 8	syl2anc 584	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
11	5, 10	eqtrd 2769	. . 3 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → (1st ‘𝑝) = 𝐶)
12		fvexd 6888	. . . 4 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2nd ‘𝑝) ∈ V)
13		simplrl 776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → 𝑝 = ⟨𝐶, 𝐷⟩)
14	13	fveq2d 6877	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2nd ‘𝑝) = (2nd ‘⟨𝐶, 𝐷⟩))
15		op2ndg 7996	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
16	6, 7, 15	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
17	16	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
18	14, 17	eqtrd 2769	. . . 4 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2nd ‘𝑝) = 𝐷)
19		simplr 768	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
20		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
21	19, 20	oveq12d 7418	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
22		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸))
23	22	simprd 495	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑒 = 𝐸)
24	19, 23	oveq12d 7418	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑒) = (𝐶 Func 𝐸))
25	20, 23	oveq12d 7418	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑑 FuncCat 𝑒) = (𝐷 FuncCat 𝐸))
26	25	fveq2d 6877	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (oppCat‘(𝑑 FuncCat 𝑒)) = (oppCat‘(𝐷 FuncCat 𝐸)))
27		ranfval.o	. . . . . . . . 9 ⊢ 𝑂 = (oppCat‘𝑅)
28		lanfval.r	. . . . . . . . . 10 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
29	28	fveq2i 6876	. . . . . . . . 9 ⊢ (oppCat‘𝑅) = (oppCat‘(𝐷 FuncCat 𝐸))
30	27, 29	eqtri 2757	. . . . . . . 8 ⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))
31	26, 30	eqtr4di 2787	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (oppCat‘(𝑑 FuncCat 𝑒)) = 𝑂)
32	19, 23	oveq12d 7418	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑐 FuncCat 𝑒) = (𝐶 FuncCat 𝐸))
33	32	fveq2d 6877	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (oppCat‘(𝑐 FuncCat 𝑒)) = (oppCat‘(𝐶 FuncCat 𝐸)))
34		ranfval.p	. . . . . . . . 9 ⊢ 𝑃 = (oppCat‘𝑆)
35		lanfval.s	. . . . . . . . . 10 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
36	35	fveq2i 6876	. . . . . . . . 9 ⊢ (oppCat‘𝑆) = (oppCat‘(𝐶 FuncCat 𝐸))
37	34, 36	eqtri 2757	. . . . . . . 8 ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))
38	33, 37	eqtr4di 2787	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (oppCat‘(𝑐 FuncCat 𝑒)) = 𝑃)
39	31, 38	oveq12d 7418	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((oppCat‘(𝑑 FuncCat 𝑒))UP(oppCat‘(𝑐 FuncCat 𝑒))) = (𝑂UP𝑃))
40	20, 23	opeq12d 4855	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ⟨𝑑, 𝑒⟩ = ⟨𝐷, 𝐸⟩)
41	40	fvoveq1d 7422	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (oppFunc‘(⟨𝑑, 𝑒⟩ −∘F 𝑓)) = (oppFunc‘(⟨𝐷, 𝐸⟩ −∘F 𝑓)))
42		eqidd 2735	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑥 = 𝑥)
43	39, 41, 42	oveq123d 7421	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((oppFunc‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒))UP(oppCat‘(𝑐 FuncCat 𝑒)))𝑥) = ((oppFunc‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂UP𝑃)𝑥))
44	21, 24, 43	mpoeq123dv 7477	. . . 4 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((oppFunc‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒))UP(oppCat‘(𝑐 FuncCat 𝑒)))𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((oppFunc‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂UP𝑃)𝑥)))
45	12, 18, 44	csbied2 3909	. . 3 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 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)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((oppFunc‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂UP𝑃)𝑥)))
46	3, 11, 45	csbied2 3909	. 2 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → ⦋(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)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((oppFunc‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂UP𝑃)𝑥)))
47	6	elexd 3481	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
48	7	elexd 3481	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
49	47, 48	opelxpd 5691	. 2 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (V × V))
50		lanfval.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
51	50	elexd 3481	. 2 ⊢ (𝜑 → 𝐸 ∈ V)
52		ovex 7433	. . . 4 ⊢ (𝐶 Func 𝐷) ∈ V
53		ovex 7433	. . . 4 ⊢ (𝐶 Func 𝐸) ∈ V
54	52, 53	mpoex 8073	. . 3 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((oppFunc‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂UP𝑃)𝑥)) ∈ V
55	54	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((oppFunc‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂UP𝑃)𝑥)) ∈ V)
56	2, 46, 49, 51, 55	ovmpod 7554	1 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩Ran𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((oppFunc‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂UP𝑃)𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3457  ⦋csb 3872  ⟨cop 4605   × cxp 5650  ‘cfv 6528  (class class class)co 7400   ∈ cmpo 7402  1^st c1st 7981  2^nd c2nd 7982  oppCatcoppc 17710   Func cfunc 17854   FuncCat cfuc 17945  oppFunccoppf 48950  UPcup 48974   −∘_F cprcof 49147  Rancran 49344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-ran 49346

This theorem is referenced by:  reldmran2  49354  ranval  49356  ranrcl  49358