Theorem ntrneik13 44047

Description: The interior of the intersection of any pair equals intersection of interiors if and only if the intersection of any pair belonging to the neighborhood of a point is equivalent to both of the pair belonging to the neighborhood 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
ntrneik13	⊢ (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 ntrneik13

Step	Hyp	Ref	Expression
1		dfss3 3945	. . . . . . . . 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 44025	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
6		elmapi 8857	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7	5, 6	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
8	7	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
9	2, 3, 4	ntrneibex 44022	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ V)
10	9	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
11		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
12		elpwi 4580	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
13		ssinss1 4219	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝐵 → (𝑠 ∩ 𝑡) ⊆ 𝐵)
14	11, 12, 13	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠 ∩ 𝑡) ⊆ 𝐵)
15	10, 14	sselpwd 5295	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠 ∩ 𝑡) ∈ 𝒫 𝐵)
16	8, 15	ffvelcdmd 7071	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠 ∩ 𝑡)) ∈ 𝒫 𝐵)
17	16	elpwid 4582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠 ∩ 𝑡)) ⊆ 𝐵)
18		ralss 4031	. . . . . . . . . 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)))))
19	17, 18	syl 17	. . . . . . . . 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)))))
20	1, 19	bitrid 283	. . . . . . . 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)))))
21		dfss3 3945	. . . . . . . . 9 ⊢ (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ∀𝑥 ∈ ((𝐼‘𝑠) ∩ (𝐼‘𝑡))𝑥 ∈ (𝐼‘(𝑠 ∩ 𝑡)))
22	7	ffvelcdmda 7070	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
23	22	elpwid 4582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
24		ssinss1 4219	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝑠) ⊆ 𝐵 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ 𝐵)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ 𝐵)
26	25	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ 𝐵)
27		ralss 4031	. . . . . . . . . 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)))))
28	26, 27	syl 17	. . . . . . . . 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)))))
29	21, 28	bitrid 283	. . . . . . . 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)))))
30	20, 29	anbi12d 632	. . . . . . 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))))))
31		eqss 3972	. . . . . . 7 ⊢ ((𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ((𝐼‘(𝑠 ∩ 𝑡)) ⊆ ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ∧ ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡))))
32		ralbiim 3099	. . . . . . 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)))))
33	30, 31, 32	3bitr4g 314	. . . . . 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)))))
34	4	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
35		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
36	9	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
37		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
38	37	elpwid 4582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ⊆ 𝐵)
39	38, 13	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑠 ∩ 𝑡) ⊆ 𝐵)
40	36, 39	sselpwd 5295	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑠 ∩ 𝑡) ∈ 𝒫 𝐵)
41	40	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑠 ∩ 𝑡) ∈ 𝒫 𝐵)
42	2, 3, 34, 35, 41	ntrneiel 44030	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘(𝑠 ∩ 𝑡)) ↔ (𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥)))
43		elin 3940	. . . . . . . . 9 ⊢ (𝑥 ∈ ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ (𝑥 ∈ (𝐼‘𝑠) ∧ 𝑥 ∈ (𝐼‘𝑡)))
44		simpllr 775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ∈ 𝒫 𝐵)
45	2, 3, 34, 35, 44	ntrneiel 44030	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑥)))
46		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑡 ∈ 𝒫 𝐵)
47	2, 3, 34, 35, 46	ntrneiel 44030	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑡) ↔ 𝑡 ∈ (𝑁‘𝑥)))
48	45, 47	anbi12d 632	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ (𝐼‘𝑠) ∧ 𝑥 ∈ (𝐼‘𝑡)) ↔ (𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥))))
49	43, 48	bitrid 283	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ (𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥))))
50	42, 49	bibi12d 345	. . . . . . 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 3159	. . . . . 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	33, 51	bitrd 279	. . . . 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 3159	. . . 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 3268	. . . 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 287	. . 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 3159	. 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 3268	. 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 287	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)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  {crab 3413  Vcvv 3457   ∩ cin 3923   ⊆ wss 3924  𝒫 cpw 4573   class class class wbr 5116   ↦ cmpt 5198  ⟶wf 6523  ‘cfv 6527  (class class class)co 7399   ∈ cmpo 7401   ↑_m cmap 8834

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 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

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 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7402  df-oprab 7403  df-mpo 7404  df-1st 7982  df-2nd 7983  df-map 8836

This theorem is referenced by: (None)