Theorem ntrneix13 45289

Description: The closure of the union of any pair is equal to the union of closures if and only if the union of any pair belonging to the convergents of a point if equivalent to at least one of the pain belonging to the convergents of that point. (Contributed by RP, 19-Jun-2021.)

Hypotheses

Ref	Expression
ntrnei.o	⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
ntrnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r	⊢ (𝜑 → 𝐼𝐹𝑁)

Assertion

Ref	Expression
ntrneix13	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠,𝑡,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠,𝑡,𝑥

Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥, 𝑡, 𝑖, 𝑗, 𝑘, 𝑚, 𝑠, 𝑙)   𝐼(𝑡, 𝑖, 𝑗, 𝑠)   𝑁(𝑥, 𝑡, 𝑖, 𝑗, 𝑘, 𝑚, 𝑠, 𝑙)   𝑂(𝑥, 𝑡, 𝑖, 𝑗, 𝑘, 𝑚, 𝑠, 𝑙)

Proof of Theorem ntrneix13

Step	Hyp	Ref	Expression
1		dfss3 3902	. . . . . . . . 9 ⊢ ((𝐼‘(𝑠 ∪ 𝑡)) ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡))𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))
2		ntrnei.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
3		ntrnei.f	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
4		ntrnei.r	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼𝐹𝑁)
5	2, 3, 4	ntrneiiex 45266	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
6	5	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
7		elmapi 8894	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
8	6, 7	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
9	2, 3, 4	ntrneibex 45263	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ V)
10	9	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
11		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
12	11	elpwid 4569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ⊆ 𝐵)
13		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
14	13	elpwid 4569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 ⊆ 𝐵)
15	12, 14	unssd 4193	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠 ∪ 𝑡) ⊆ 𝐵)
16	10, 15	sselpwd 5294	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝐵)
17	8, 16	ffvelcdmd 7106	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠 ∪ 𝑡)) ∈ 𝒫 𝐵)
18	17	elpwid 4569	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠 ∪ 𝑡)) ⊆ 𝐵)
19		ralss 3986	. . . . . . . . . 10 ⊢ ((𝐼‘(𝑠 ∪ 𝑡)) ⊆ 𝐵 → (∀𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡))𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) → 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))))
20	18, 19	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡))𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) → 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))))
21	1, 20	bitrid 286	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠 ∪ 𝑡)) ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) → 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))))
22		dfss3 3902	. . . . . . . . 9 ⊢ (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∪ 𝑡)) ↔ ∀𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡))𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)))
23	8, 11	ffvelcdmd 7106	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
24	23	elpwid 4569	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
25	8, 13	ffvelcdmd 7106	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ∈ 𝒫 𝐵)
26	25	elpwid 4569	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ⊆ 𝐵)
27	24, 26	unssd 4193	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ 𝐵)
28		ralss 3986	. . . . . . . . . 10 ⊢ (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ 𝐵 → (∀𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡))𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) → 𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)))))
29	27, 28	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡))𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) → 𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)))))
30	22, 29	bitrid 286	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∪ 𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) → 𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)))))
31	21, 30	anbi12d 644	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼‘(𝑠 ∪ 𝑡)) ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ∧ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∪ 𝑡))) ↔ (∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) → 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡))) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) → 𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡))))))
32		eqss 3928	. . . . . . 7 ⊢ ((𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ((𝐼‘(𝑠 ∪ 𝑡)) ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ∧ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∪ 𝑡))))
33		ralbiim 3128	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) ↔ 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡))) ↔ (∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) → 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡))) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) → 𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)))))
34	31, 32, 33	3bitr4g 317	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) ↔ 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))))
35	4	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
36		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
37	9	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ V)
38		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ∈ 𝒫 𝐵)
39	38	elpwid 4569	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ⊆ 𝐵)
40		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑡 ∈ 𝒫 𝐵)
41	40	elpwid 4569	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑡 ⊆ 𝐵)
42	39, 41	unssd 4193	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑠 ∪ 𝑡) ⊆ 𝐵)
43	37, 42	sselpwd 5294	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝐵)
44	2, 3, 35, 36, 43	ntrneiel 45271	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) ↔ (𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥)))
45		elun 4155	. . . . . . . . 9 ⊢ (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)))
46	2, 3, 35, 36, 38	ntrneiel 45271	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑥)))
47	2, 3, 35, 36, 40	ntrneiel 45271	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑡) ↔ 𝑡 ∈ (𝑁‘𝑥)))
48	46, 47	orbi12d 932	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))))
49	45, 48	bitrid 286	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))))
50	44, 49	bibi12d 348	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) ↔ 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡))) ↔ ((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
51	50	ralbidva 3187	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘(𝑠 ∪ 𝑡)) ↔ 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡))) ↔ ∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
52	34, 51	bitrd 282	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
53	52	ralbidva 3187	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
54		ralcom 3294	. . . 4 ⊢ (∀𝑡 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))))
55	53, 54	bitrdi 290	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
56	55	ralbidva 3187	. 2 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
57		ralcom 3294	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))))
58	56, 57	bitrdi 290	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  𝒫 cpw 4560   class class class wbr 5106   ↦ cmpt 5189  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ∈ cmpo 7442   ↑_m cmap 8872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-map 8874

This theorem is used by: (None)