ntrclsk3 - Mathbox for Richard Penner

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Theorem ntrclsk3 41266

Description: The intersection of interiors of a every pair is a subset of the interior of the intersection of the pair if an only if the closure of the union of every pair is a subset of the union of closures of the pair. (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
ntrclsk3	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 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 ntrclsk3 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6686	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐼‘𝑠) = (𝐼‘𝑎))
2	1	ineq1d 4112	. . . 4 ⊢ (𝑠 = 𝑎 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑡)))
3		ineq1 4106	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝑠 ∩ 𝑡) = (𝑎 ∩ 𝑡))
4	3	fveq2d 6690	. . . 4 ⊢ (𝑠 = 𝑎 → (𝐼‘(𝑠 ∩ 𝑡)) = (𝐼‘(𝑎 ∩ 𝑡)))
5	2, 4	sseq12d 3920	. . 3 ⊢ (𝑠 = 𝑎 → (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ((𝐼‘𝑎) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑎 ∩ 𝑡))))
6		fveq2 6686	. . . . 5 ⊢ (𝑡 = 𝑏 → (𝐼‘𝑡) = (𝐼‘𝑏))
7	6	ineq2d 4113	. . . 4 ⊢ (𝑡 = 𝑏 → ((𝐼‘𝑎) ∩ (𝐼‘𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)))
8		ineq2 4107	. . . . 5 ⊢ (𝑡 = 𝑏 → (𝑎 ∩ 𝑡) = (𝑎 ∩ 𝑏))
9	8	fveq2d 6690	. . . 4 ⊢ (𝑡 = 𝑏 → (𝐼‘(𝑎 ∩ 𝑡)) = (𝐼‘(𝑎 ∩ 𝑏)))
10	7, 9	sseq12d 3920	. . 3 ⊢ (𝑡 = 𝑏 → (((𝐼‘𝑎) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑎 ∩ 𝑡)) ↔ ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘(𝑎 ∩ 𝑏))))
11	5, 10	cbvral2vw 3363	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ∀𝑎 ∈ 𝒫 𝐵∀𝑏 ∈ 𝒫 𝐵((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘(𝑎 ∩ 𝑏)))
12		ntrcls.d	. . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵)
13		ntrcls.r	. . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾)
14	12, 13	ntrclsbex 41230	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
15		difssd 4033	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ⊆ 𝐵)
16	14, 15	sselpwd 5204	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
17	16	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
18		elpwi 4507	. . . 4 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
19		simpl 486	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → 𝐵 ∈ V)
20		difssd 4033	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ⊆ 𝐵)
21	19, 20	sselpwd 5204	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ∈ 𝒫 𝐵)
22		simpr 488	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → 𝑠 = (𝐵 ∖ 𝑎))
23	22	difeq2d 4023	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑎)))
24	23	eqeq2d 2750	. . . . . 6 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎))))
25		eqcom 2746	. . . . . 6 ⊢ (𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
26	24, 25	bitrdi 290	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎))
27		dfss4 4159	. . . . . . 7 ⊢ (𝑎 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
28	27	biimpi 219	. . . . . 6 ⊢ (𝑎 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
29	28	adantl 485	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
30	21, 26, 29	rspcedvd 3532	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
31	14, 18, 30	syl2an 599	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
32		simpl1 1192	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
33		difssd 4033	. . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ⊆ 𝐵)
34	14, 33	sselpwd 5204	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
35	32, 34	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
36		elpwi 4507	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵)
37		simpl 486	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → 𝐵 ∈ V)
38		difssd 4033	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ⊆ 𝐵)
39	37, 38	sselpwd 5204	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ∈ 𝒫 𝐵)
40		simpr 488	. . . . . . . . . 10 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → 𝑡 = (𝐵 ∖ 𝑏))
41	40	difeq2d 4023	. . . . . . . . 9 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝐵 ∖ 𝑡) = (𝐵 ∖ (𝐵 ∖ 𝑏)))
42	41	eqeq2d 2750	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏))))
43		eqcom 2746	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
44	42, 43	bitrdi 290	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏))
45		dfss4 4159	. . . . . . . . 9 ⊢ (𝑏 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
46	45	biimpi 219	. . . . . . . 8 ⊢ (𝑏 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
47	46	adantl 485	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
48	39, 44, 47	rspcedvd 3532	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
49	14, 36, 48	syl2an 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
50	49	3ad2antl1 1186	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
51		simp13 1206	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑎 = (𝐵 ∖ 𝑠))
52		fveq2 6686	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝐼‘𝑎) = (𝐼‘(𝐵 ∖ 𝑠)))
53	52	ineq1d 4112	. . . . . . 7 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)))
54		ineq1 4106	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝑎 ∩ 𝑏) = ((𝐵 ∖ 𝑠) ∩ 𝑏))
55	54	fveq2d 6690	. . . . . . 7 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝐼‘(𝑎 ∩ 𝑏)) = (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)))
56	53, 55	sseq12d 3920	. . . . . 6 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘(𝑎 ∩ 𝑏)) ↔ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏))))
57	51, 56	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘(𝑎 ∩ 𝑏)) ↔ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏))))
58		fveq2 6686	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → (𝐼‘𝑏) = (𝐼‘(𝐵 ∖ 𝑡)))
59	58	ineq2d 4113	. . . . . . 7 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))))
60		ineq2 4107	. . . . . . . . 9 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐵 ∖ 𝑠) ∩ 𝑏) = ((𝐵 ∖ 𝑠) ∩ (𝐵 ∖ 𝑡)))
61		difundi 4180	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝑠 ∪ 𝑡)) = ((𝐵 ∖ 𝑠) ∩ (𝐵 ∖ 𝑡))
62	60, 61	eqtr4di 2792	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐵 ∖ 𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠 ∪ 𝑡)))
63	62	fveq2d 6690	. . . . . . 7 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))))
64	59, 63	sseq12d 3920	. . . . . 6 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → (((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) ↔ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))))
65	64	3ad2ant3 1136	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) ↔ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))))
66		simp11 1204	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝜑)
67		ntrcls.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
68	67, 12, 13	ntrclsiex 41249	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
69	68, 14	jca 515	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V))
70	66, 69	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V))
71		elmapi 8471	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
72	71	adantr 484	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
73		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
74		difssd 4033	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
75	73, 74	sselpwd 5204	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
76	72, 75	ffvelrnd 6874	. . . . . . . . . 10 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
77	76	elpwid 4509	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
78		orc 866	. . . . . . . . 9 ⊢ ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵 ∖ 𝑡)) ⊆ 𝐵))
79		inss 4139	. . . . . . . . 9 ⊢ (((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵 ∖ 𝑡)) ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵)
80	77, 78, 79	3syl 18	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵)
81		difssd 4033	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠 ∪ 𝑡)) ⊆ 𝐵)
82	73, 81	sselpwd 5204	. . . . . . . . . 10 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠 ∪ 𝑡)) ∈ 𝒫 𝐵)
83	72, 82	ffvelrnd 6874	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ∈ 𝒫 𝐵)
84	83	elpwid 4509	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵)
85	80, 84	jca 515	. . . . . . 7 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵))
86		sscon34b 4195	. . . . . . 7 ⊢ ((((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵) → (((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))))))
87	70, 85, 86	3syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))))))
88		difindi 4182	. . . . . . . 8 ⊢ (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
89	88	sseq2i 3916	. . . . . . 7 ⊢ ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡)))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡)))))
90	89	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) ⊆ (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡)))) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))))
91	66, 14	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝐵 ∈ V)
92	66, 68	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
93		simp12 1205	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑠 ∈ 𝒫 𝐵)
94		rp-simp2 40987	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑡 ∈ 𝒫 𝐵)
95		simpl2 1193	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
96		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
97		eqid 2739	. . . . . . . . . 10 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
98		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
99		simprl 771	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
100	99	elpwid 4509	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ⊆ 𝐵)
101		simprr 773	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
102	101	elpwid 4509	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ⊆ 𝐵)
103	100, 102	unssd 4086	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ⊆ 𝐵)
104	98, 103	sselpwd 5204	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝐵)
105	104	3ad2antl2 1187	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝐵)
106		eqid 2739	. . . . . . . . . 10 ⊢ ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡))
107	67, 12, 95, 96, 97, 105, 106	dssmapfv3d 41213	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))))
108		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝜑)
109	67, 12, 13	ntrclsfv1 41251	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
110	109	fveq1d 6688	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = (𝐾‘(𝑠 ∪ 𝑡)))
111	108, 110	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = (𝐾‘(𝑠 ∪ 𝑡)))
112	107, 111	eqtr3d 2776	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) = (𝐾‘(𝑠 ∪ 𝑡)))
113		simprl 771	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
114		eqid 2739	. . . . . . . . . . 11 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
115	67, 12, 95, 96, 97, 113, 114	dssmapfv3d 41213	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
116	109	fveq1d 6688	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
117	108, 116	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
118	115, 117	eqtr3d 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) = (𝐾‘𝑠))
119		simprr 773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
120		eqid 2739	. . . . . . . . . . 11 ⊢ ((𝐷‘𝐼)‘𝑡) = ((𝐷‘𝐼)‘𝑡)
121	67, 12, 95, 96, 97, 119, 120	dssmapfv3d 41213	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
122	109	fveq1d 6688	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐷‘𝐼)‘𝑡) = (𝐾‘𝑡))
123	108, 122	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑡) = (𝐾‘𝑡))
124	121, 123	eqtr3d 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))) = (𝐾‘𝑡))
125	118, 124	uneq12d 4064	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡)))) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡)))
126	112, 125	sseq12d 3920	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡)))) ↔ (𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
127	66, 91, 92, 93, 94, 126	syl32anc 1379	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) ⊆ ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡)))) ↔ (𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
128	87, 90, 127	3bitrd 308	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ↔ (𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
129	57, 65, 128	3bitrd 308	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘(𝑎 ∩ 𝑏)) ↔ (𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
130	35, 50, 129	ralxfrd2 5289	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) → (∀𝑏 ∈ 𝒫 𝐵((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘(𝑎 ∩ 𝑏)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
131	17, 31, 130	ralxfrd2 5289	. 2 ⊢ (𝜑 → (∀𝑎 ∈ 𝒫 𝐵∀𝑏 ∈ 𝒫 𝐵((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘(𝑎 ∩ 𝑏)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
132	11, 131	syl5bb 286	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 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 209 ∧ wa 399 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∃wrex 3055 Vcvv 3400 ∖ cdif 3850 ∪ cun 3851 ∩ cin 3852 ⊆ wss 3853 𝒫 cpw 4498 class class class wbr 5040 ↦ cmpt 5120 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 ↑_m cmap 8449

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-frege1 40984

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7185 df-oprab 7186 df-mpo 7187 df-1st 7726 df-2nd 7727 df-map 8451

This theorem is referenced by: (None)

W3C validator