Theorem upfval2 49208

Description: Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.)

Hypotheses

Ref	Expression
upfval.b	⊢ 𝐵 = (Base‘𝐷)
upfval.c	⊢ 𝐶 = (Base‘𝐸)
upfval.h	⊢ 𝐻 = (Hom ‘𝐷)
upfval.j	⊢ 𝐽 = (Hom ‘𝐸)
upfval.o	⊢ 𝑂 = (comp‘𝐸)
upfval2.w	⊢ (𝜑 → 𝑊 ∈ 𝐶)
upfval2.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))

Assertion

Ref	Expression
upfval2	⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))})

Distinct variable groups:   𝐵,𝑔,𝑘,𝑚,𝑥,𝑦   𝐶,𝑔,𝑘,𝑚,𝑥,𝑦   𝐷,𝑔,𝑘,𝑚,𝑥,𝑦   𝑔,𝐸,𝑘,𝑚,𝑥,𝑦   𝑔,𝐹,𝑘,𝑚,𝑥,𝑦   𝑔,𝐻,𝑘,𝑚,𝑥,𝑦   𝑔,𝐽,𝑘,𝑚,𝑥,𝑦   𝑔,𝑂,𝑘,𝑚,𝑥,𝑦   𝑔,𝑊,𝑘,𝑚,𝑥,𝑦   𝜑,𝑚,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑔,𝑘)

Proof of Theorem upfval2

Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upfval2.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
2		upfval2.w	. 2 ⊢ (𝜑 → 𝑊 ∈ 𝐶)
3		anass 468	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))))
4	3	opabbii 5158	. . 3 ⊢ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ (𝑥 ∈ 𝐵 ∧ (𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))}
5		upfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
6	5	fvexi 6836	. . . . 5 ⊢ 𝐵 ∈ V
7	6	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
8		simprl 770	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))) → 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥)))
9		ovexd 7381	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑊𝐽((1st ‘𝐹)‘𝑥)) ∈ V)
10	8, 9	abexd 5263	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → {𝑚 ∣ (𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))} ∈ V)
11	7, 10	opabex3d 7897	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑚⟩ ∣ (𝑥 ∈ 𝐵 ∧ (𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))} ∈ V)
12	4, 11	eqeltrid 2835	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))} ∈ V)
13		fveq2 6822	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (1st ‘𝑓) = (1st ‘𝐹))
14	13	fveq1d 6824	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑥))
15	14	oveq2d 7362	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑤𝐽((1st ‘𝑓)‘𝑥)) = (𝑤𝐽((1st ‘𝐹)‘𝑥)))
16	15	eleq2d 2817	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥)) ↔ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥))))
17	16	anbi2d 630	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥)))))
18	13	fveq1d 6824	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((1st ‘𝑓)‘𝑦) = ((1st ‘𝐹)‘𝑦))
19	18	oveq2d 7362	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑤𝐽((1st ‘𝑓)‘𝑦)) = (𝑤𝐽((1st ‘𝐹)‘𝑦)))
20	14	opeq2d 4832	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩ = ⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩)
21	20, 18	oveq12d 7364	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦)) = (⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦)))
22		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (2nd ‘𝑓) = (2nd ‘𝐹))
23	22	oveqd 7363	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑥(2nd ‘𝑓)𝑦) = (𝑥(2nd ‘𝐹)𝑦))
24	23	fveq1d 6824	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((𝑥(2nd ‘𝑓)𝑦)‘𝑘) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑘))
25		eqidd 2732	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → 𝑚 = 𝑚)
26	21, 24, 25	oveq123d 7367	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚) = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))
27	26	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚) ↔ 𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))
28	27	reubidv 3362	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))
29	19, 28	raleqbidv 3312	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑔 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))
30	29	ralbidv 3155	. . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))
31	17, 30	anbi12d 632	. . . 4 ⊢ (𝑓 = 𝐹 → (((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))))
32	31	opabbidv 5157	. . 3 ⊢ (𝑓 = 𝐹 → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))})
33		oveq1 7353	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑤𝐽((1st ‘𝐹)‘𝑥)) = (𝑊𝐽((1st ‘𝐹)‘𝑥)))
34	33	eleq2d 2817	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥)) ↔ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))))
35	34	anbi2d 630	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥)))))
36		oveq1 7353	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑤𝐽((1st ‘𝐹)‘𝑦)) = (𝑊𝐽((1st ‘𝐹)‘𝑦)))
37		opeq1 4825	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩ = ⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩)
38	37	oveq1d 7361	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦)) = (⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦)))
39	38	oveqd 7363	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚) = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))
40	39	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚) ↔ 𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))
41	40	reubidv 3362	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))
42	36, 41	raleqbidv 3312	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑔 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚) ↔ ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))
43	42	ralbidv 3155	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)))
44	35, 43	anbi12d 632	. . . 4 ⊢ (𝑤 = 𝑊 → (((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))))
45	44	opabbidv 5157	. . 3 ⊢ (𝑤 = 𝑊 → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))})
46		upfval.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
47		upfval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
48		upfval.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
49		upfval.o	. . . 4 ⊢ 𝑂 = (comp‘𝐸)
50	5, 46, 47, 48, 49	upfval 49207	. . 3 ⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})
51	32, 45, 50	ovmpog 7505	. 2 ⊢ ((𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ 𝐶 ∧ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))} ∈ V) → (𝐹(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))})
52	1, 2, 12, 51	syl3anc 1373	1 ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃!wreu 3344  Vcvv 3436  ⟨cop 4582  {copab 5153  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17117  Hom chom 17169  compcco 17170   Func cfunc 17758   UP cup 49204

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-func 17762  df-up 49205

This theorem is referenced by:  upfval3  49209