ntrclsk13 - Mathbox for Richard Penner

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Theorem ntrclsk13 42807

Description: The interior of the intersection of any pair is equal to the intersection of the interiors if and only if the closure of the unions of any pair is equal to the union of closures. (Contributed by RP, 19-Jun-2021.)

**Hypotheses**
Ref	Expression
ntrcls.o	⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)

**Assertion**
Ref	Expression
ntrclsk13	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))

Distinct variable groups: 𝐵,𝑠,𝑡,𝑖,𝑗,𝑘 𝐼,𝑠,𝑡,𝑖,𝑗,𝑘 𝜑,𝑠,𝑡,𝑖,𝑗,𝑘 Allowed substitution hints: 𝐷(𝑡,𝑖,𝑗,𝑘,𝑠) 𝐾(𝑡,𝑖,𝑗,𝑘,𝑠) 𝑂(𝑡,𝑖,𝑗,𝑘,𝑠)

Proof of Theorem ntrclsk13 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 4204	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝑠 ∩ 𝑡) = (𝑎 ∩ 𝑡))
2	1	fveq2d 6892	. . . 4 ⊢ (𝑠 = 𝑎 → (𝐼‘(𝑠 ∩ 𝑡)) = (𝐼‘(𝑎 ∩ 𝑡)))
3		fveq2 6888	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐼‘𝑠) = (𝐼‘𝑎))
4	3	ineq1d 4210	. . . 4 ⊢ (𝑠 = 𝑎 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑡)))
5	2, 4	eqeq12d 2748	. . 3 ⊢ (𝑠 = 𝑎 → ((𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ (𝐼‘(𝑎 ∩ 𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑡))))
6		ineq2 4205	. . . . 5 ⊢ (𝑡 = 𝑏 → (𝑎 ∩ 𝑡) = (𝑎 ∩ 𝑏))
7	6	fveq2d 6892	. . . 4 ⊢ (𝑡 = 𝑏 → (𝐼‘(𝑎 ∩ 𝑡)) = (𝐼‘(𝑎 ∩ 𝑏)))
8		fveq2 6888	. . . . 5 ⊢ (𝑡 = 𝑏 → (𝐼‘𝑡) = (𝐼‘𝑏))
9	8	ineq2d 4211	. . . 4 ⊢ (𝑡 = 𝑏 → ((𝐼‘𝑎) ∩ (𝐼‘𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)))
10	7, 9	eqeq12d 2748	. . 3 ⊢ (𝑡 = 𝑏 → ((𝐼‘(𝑎 ∩ 𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑡)) ↔ (𝐼‘(𝑎 ∩ 𝑏)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏))))
11	5, 10	cbvral2vw 3238	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ∀𝑎 ∈ 𝒫 𝐵∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎 ∩ 𝑏)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)))
12		ntrcls.d	. . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵)
13		ntrcls.r	. . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾)
14	12, 13	ntrclsbex 42770	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
15		difssd 4131	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ⊆ 𝐵)
16	14, 15	sselpwd 5325	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
17	16	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
18		elpwi 4608	. . . 4 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
19	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐵) → 𝐵 ∈ V)
20		difssd 4131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ⊆ 𝐵)
21	19, 20	sselpwd 5325	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ∈ 𝒫 𝐵)
22		difeq2 4115	. . . . . . . 8 ⊢ (𝑠 = (𝐵 ∖ 𝑎) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑎)))
23	22	eqeq2d 2743	. . . . . . 7 ⊢ (𝑠 = (𝐵 ∖ 𝑎) → (𝑎 = (𝐵 ∖ 𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎))))
24		eqcom 2739	. . . . . . 7 ⊢ (𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
25	23, 24	bitrdi 286	. . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑎) → (𝑎 = (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎))
26	25	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎))
27		dfss4 4257	. . . . . . 7 ⊢ (𝑎 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
28	27	biimpi 215	. . . . . 6 ⊢ (𝑎 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
29	28	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
30	21, 26, 29	rspcedvd 3614	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
31	18, 30	sylan2 593	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
32		ineq1 4204	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝑎 ∩ 𝑏) = ((𝐵 ∖ 𝑠) ∩ 𝑏))
33	32	fveq2d 6892	. . . . . . 7 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝐼‘(𝑎 ∩ 𝑏)) = (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)))
34		fveq2 6888	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝐼‘𝑎) = (𝐼‘(𝐵 ∖ 𝑠)))
35	34	ineq1d 4210	. . . . . . 7 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)))
36	33, 35	eqeq12d 2748	. . . . . 6 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → ((𝐼‘(𝑎 ∩ 𝑏)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ↔ (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏))))
37	36	ralbidv 3177	. . . . 5 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎 ∩ 𝑏)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ↔ ∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏))))
38	37	3ad2ant3 1135	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎 ∩ 𝑏)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ↔ ∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏))))
39		difssd 4131	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ⊆ 𝐵)
40	14, 39	sselpwd 5325	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
41	40	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
42		simpll 765	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → 𝜑)
43		elpwi 4608	. . . . . . . 8 ⊢ (𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵)
44	43	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → 𝑏 ⊆ 𝐵)
45		difssd 4131	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∖ 𝑏) ⊆ 𝐵)
46	14, 45	sselpwd 5325	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ∖ 𝑏) ∈ 𝒫 𝐵)
47	46	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ∈ 𝒫 𝐵)
48		difeq2 4115	. . . . . . . . . . 11 ⊢ (𝑡 = (𝐵 ∖ 𝑏) → (𝐵 ∖ 𝑡) = (𝐵 ∖ (𝐵 ∖ 𝑏)))
49	48	eqeq2d 2743	. . . . . . . . . 10 ⊢ (𝑡 = (𝐵 ∖ 𝑏) → (𝑏 = (𝐵 ∖ 𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏))))
50		eqcom 2739	. . . . . . . . . 10 ⊢ (𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
51	49, 50	bitrdi 286	. . . . . . . . 9 ⊢ (𝑡 = (𝐵 ∖ 𝑏) → (𝑏 = (𝐵 ∖ 𝑡) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏))
52	51	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏))
53		dfss4 4257	. . . . . . . . . 10 ⊢ (𝑏 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
54	53	biimpi 215	. . . . . . . . 9 ⊢ (𝑏 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
55	54	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
56	47, 52, 55	rspcedvd 3614	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
57	42, 44, 56	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
58		ineq2 4205	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐵 ∖ 𝑠) ∩ 𝑏) = ((𝐵 ∖ 𝑠) ∩ (𝐵 ∖ 𝑡)))
59		difundi 4278	. . . . . . . . . . 11 ⊢ (𝐵 ∖ (𝑠 ∪ 𝑡)) = ((𝐵 ∖ 𝑠) ∩ (𝐵 ∖ 𝑡))
60	58, 59	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐵 ∖ 𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠 ∪ 𝑡)))
61	60	fveq2d 6892	. . . . . . . . 9 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))))
62		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → (𝐼‘𝑏) = (𝐼‘(𝐵 ∖ 𝑡)))
63	62	ineq2d 4211	. . . . . . . . 9 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))))
64	61, 63	eqeq12d 2748	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) ↔ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡)))))
65	64	3ad2ant3 1135	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) ↔ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡)))))
66		simp1l 1197	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝜑)
67	66, 14	jccir 522	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝜑 ∧ 𝐵 ∈ V))
68		simp1r 1198	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑠 ∈ 𝒫 𝐵)
69		simp2 1137	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑡 ∈ 𝒫 𝐵)
70		ntrcls.o	. . . . . . . . . . . . . 14 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
71	70, 12, 13	ntrclsiex 42789	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
72		elmapi 8839	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
73	71, 72	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
74	73	anim1i 615	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V))
75	74	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V))
76		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
77		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
78		difssd 4131	. . . . . . . . . . . . . 14 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠 ∪ 𝑡)) ⊆ 𝐵)
79	77, 78	sselpwd 5325	. . . . . . . . . . . . 13 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠 ∪ 𝑡)) ∈ 𝒫 𝐵)
80	76, 79	ffvelcdmd 7084	. . . . . . . . . . . 12 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ∈ 𝒫 𝐵)
81	80	elpwid 4610	. . . . . . . . . . 11 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵)
82		difssd 4131	. . . . . . . . . . . . . . 15 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
83	77, 82	sselpwd 5325	. . . . . . . . . . . . . 14 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
84	76, 83	ffvelcdmd 7084	. . . . . . . . . . . . 13 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
85	84	elpwid 4610	. . . . . . . . . . . 12 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
86		ssinss1 4236	. . . . . . . . . . . 12 ⊢ ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵)
87	85, 86	syl 17	. . . . . . . . . . 11 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵)
88	81, 87	jca 512	. . . . . . . . . 10 ⊢ ((𝐼:𝒫 𝐵⟶𝒫 𝐵 ∧ 𝐵 ∈ V) → ((𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵 ∧ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵))
89		rcompleq 4294	. . . . . . . . . 10 ⊢ (((𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵 ∧ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) = (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))))))
90	75, 88, 89	3syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) = (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))))))
91		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
92	71	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
93		eqid 2732	. . . . . . . . . . 11 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
94		simprl 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
95	94	elpwid 4610	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ⊆ 𝐵)
96		simprr 771	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
97	96	elpwid 4610	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ⊆ 𝐵)
98	95, 97	unssd 4185	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ⊆ 𝐵)
99	91, 98	sselpwd 5325	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝐵)
100		eqid 2732	. . . . . . . . . . 11 ⊢ ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡))
101	70, 12, 91, 92, 93, 99, 100	dssmapfv3d 42755	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))))
102		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
103		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
104	71	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
105		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
106		eqid 2732	. . . . . . . . . . . . . 14 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
107	70, 12, 103, 104, 93, 105, 106	dssmapfv3d 42755	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
108	102, 107	sylan2 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
109		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
110		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
111	71	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
112		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
113		eqid 2732	. . . . . . . . . . . . . 14 ⊢ ((𝐷‘𝐼)‘𝑡) = ((𝐷‘𝐼)‘𝑡)
114	70, 12, 110, 111, 93, 112, 113	dssmapfv3d 42755	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐷‘𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
115	109, 114	sylan2 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
116	108, 115	uneq12d 4163	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (((𝐷‘𝐼)‘𝑠) ∪ ((𝐷‘𝐼)‘𝑡)) = ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡)))))
117		difindi 4280	. . . . . . . . . . 11 ⊢ (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
118	116, 117	eqtr4di 2790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (((𝐷‘𝐼)‘𝑠) ∪ ((𝐷‘𝐼)‘𝑡)) = (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡)))))
119	101, 118	eqeq12d 2748	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = (((𝐷‘𝐼)‘𝑠) ∪ ((𝐷‘𝐼)‘𝑡)) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) = (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))))))
120		simpll 765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝜑)
121	70, 12, 13	ntrclsfv1 42791	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
122		fveq1 6887	. . . . . . . . . . 11 ⊢ ((𝐷‘𝐼) = 𝐾 → ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = (𝐾‘(𝑠 ∪ 𝑡)))
123		fveq1 6887	. . . . . . . . . . . 12 ⊢ ((𝐷‘𝐼) = 𝐾 → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
124		fveq1 6887	. . . . . . . . . . . 12 ⊢ ((𝐷‘𝐼) = 𝐾 → ((𝐷‘𝐼)‘𝑡) = (𝐾‘𝑡))
125	123, 124	uneq12d 4163	. . . . . . . . . . 11 ⊢ ((𝐷‘𝐼) = 𝐾 → (((𝐷‘𝐼)‘𝑠) ∪ ((𝐷‘𝐼)‘𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡)))
126	122, 125	eqeq12d 2748	. . . . . . . . . 10 ⊢ ((𝐷‘𝐼) = 𝐾 → (((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = (((𝐷‘𝐼)‘𝑠) ∪ ((𝐷‘𝐼)‘𝑡)) ↔ (𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
127	120, 121, 126	3syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = (((𝐷‘𝐼)‘𝑠) ∪ ((𝐷‘𝐼)‘𝑡)) ↔ (𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
128	90, 119, 127	3bitr2d 306	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ↔ (𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
129	67, 68, 69, 128	syl12anc 835	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ↔ (𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
130	65, 129	bitrd 278	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) ↔ (𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
131	41, 57, 130	ralxfrd2 5409	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
132	131	3adant3 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
133	38, 132	bitrd 278	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎 ∩ 𝑏)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
134	17, 31, 133	ralxfrd2 5409	. 2 ⊢ (𝜑 → (∀𝑎 ∈ 𝒫 𝐵∀𝑏 ∈ 𝒫 𝐵(𝐼‘(𝑎 ∩ 𝑏)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
135	11, 134	bitrid 282	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-map 8818

This theorem is referenced by: (None)

W3C validator