Theorem upfval2 49983

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 473	. . . 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 5177	. . 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 6895	. . . . 5 ⊢ 𝐵 ∈ V
7	6	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
8		simprl 782	. . . . 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 7447	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑊𝐽((1st ‘𝐹)‘𝑥)) ∈ V)
10	8, 9	abexd 5295	. . . 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 7960	. . 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 2866	. 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 6881	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (1st ‘𝑓) = (1st ‘𝐹))
14	13	fveq1d 6883	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑥))
15	14	oveq2d 7428	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑤𝐽((1st ‘𝑓)‘𝑥)) = (𝑤𝐽((1st ‘𝐹)‘𝑥)))
16	15	eleq2d 2848	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥)) ↔ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥))))
17	16	anbi2d 641	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥)))))
18	13	fveq1d 6883	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((1st ‘𝑓)‘𝑦) = ((1st ‘𝐹)‘𝑦))
19	18	oveq2d 7428	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑤𝐽((1st ‘𝑓)‘𝑦)) = (𝑤𝐽((1st ‘𝐹)‘𝑦)))
20	14	opeq2d 4844	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩ = ⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩)
21	20, 18	oveq12d 7430	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦)) = (⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦)))
22		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (2nd ‘𝑓) = (2nd ‘𝐹))
23	22	oveqd 7429	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑥(2nd ‘𝑓)𝑦) = (𝑥(2nd ‘𝐹)𝑦))
24	23	fveq1d 6883	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((𝑥(2nd ‘𝑓)𝑦)‘𝑘) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑘))
25		eqidd 2763	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → 𝑚 = 𝑚)
26	21, 24, 25	oveq123d 7433	. . . . . . . . 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 2773	. . . . . . . 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 3384	. . . . . . 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 3337	. . . . . 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 3187	. . . . 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 643	. . . 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 5176	. . 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 7419	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑤𝐽((1st ‘𝐹)‘𝑥)) = (𝑊𝐽((1st ‘𝐹)‘𝑥)))
34	33	eleq2d 2848	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥)) ↔ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))))
35	34	anbi2d 641	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝐹)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥)))))
36		oveq1 7419	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑤𝐽((1st ‘𝐹)‘𝑦)) = (𝑊𝐽((1st ‘𝐹)‘𝑦)))
37		opeq1 4837	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩ = ⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩)
38	37	oveq1d 7427	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (⟨𝑤, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦)) = (⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦)))
39	38	oveqd 7429	. . . . . . . . 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 2773	. . . . . . . 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 3384	. . . . . . 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 3337	. . . . . 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 3187	. . . . 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 643	. . . 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 5176	. . 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 49982	. . 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 7571	. 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 1397	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))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃!wreu 3366  Vcvv 3454  ⟨cop 4594  {copab 5172  ‘cfv 6536  (class class class)co 7412  1^st c1st 7982  2^nd c2nd 7983  Basecbs 17275  Hom chom 17327  compcco 17328   Func cfunc 17917   UP cup 49979

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-func 17921  df-up 49980

This theorem is used by:  upfval3  49984