ntrclsk3 - Mathbox for Richard Penner

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Theorem ntrclsk3 38368

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 6191	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐼‘𝑠) = (𝐼‘𝑎))
2	1	ineq1d 3813	. . . 4 ⊢ (𝑠 = 𝑎 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑡)))
3		ineq1 3807	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝑠 ∩ 𝑡) = (𝑎 ∩ 𝑡))
4	3	fveq2d 6195	. . . 4 ⊢ (𝑠 = 𝑎 → (𝐼‘(𝑠 ∩ 𝑡)) = (𝐼‘(𝑎 ∩ 𝑡)))
5	2, 4	sseq12d 3634	. . 3 ⊢ (𝑠 = 𝑎 → (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ((𝐼‘𝑎) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑎 ∩ 𝑡))))
6		fveq2 6191	. . . . 5 ⊢ (𝑡 = 𝑏 → (𝐼‘𝑡) = (𝐼‘𝑏))
7	6	ineq2d 3814	. . . 4 ⊢ (𝑡 = 𝑏 → ((𝐼‘𝑎) ∩ (𝐼‘𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)))
8		ineq2 3808	. . . . 5 ⊢ (𝑡 = 𝑏 → (𝑎 ∩ 𝑡) = (𝑎 ∩ 𝑏))
9	8	fveq2d 6195	. . . 4 ⊢ (𝑡 = 𝑏 → (𝐼‘(𝑎 ∩ 𝑡)) = (𝐼‘(𝑎 ∩ 𝑏)))
10	7, 9	sseq12d 3634	. . 3 ⊢ (𝑡 = 𝑏 → (((𝐼‘𝑎) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑎 ∩ 𝑡)) ↔ ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘(𝑎 ∩ 𝑏))))
11	5, 10	cbvral2v 3179	. 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 38332	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
15		difssd 3738	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ⊆ 𝐵)
16	14, 15	sselpwd 4807	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
17	16	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
18		elpwi 4168	. . . 4 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
19		simpl 473	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → 𝐵 ∈ V)
20		difssd 3738	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ⊆ 𝐵)
21	19, 20	sselpwd 4807	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ∈ 𝒫 𝐵)
22		simpr 477	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → 𝑠 = (𝐵 ∖ 𝑎))
23	22	difeq2d 3728	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑎)))
24	23	eqeq2d 2632	. . . . . 6 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎))))
25		eqcom 2629	. . . . . 6 ⊢ (𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
26	24, 25	syl6bb 276	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎))
27		dfss4 3858	. . . . . . 7 ⊢ (𝑎 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
28	27	biimpi 206	. . . . . 6 ⊢ (𝑎 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
29	28	adantl 482	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
30	21, 26, 29	rspcedvd 3317	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
31	14, 18, 30	syl2an 494	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
32		simpl1 1064	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
33		difssd 3738	. . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ⊆ 𝐵)
34	14, 33	sselpwd 4807	. . . . 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 4168	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵)
37		simpl 473	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → 𝐵 ∈ V)
38		difssd 3738	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ⊆ 𝐵)
39	37, 38	sselpwd 4807	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ∈ 𝒫 𝐵)
40		simpr 477	. . . . . . . . . 10 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → 𝑡 = (𝐵 ∖ 𝑏))
41	40	difeq2d 3728	. . . . . . . . 9 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝐵 ∖ 𝑡) = (𝐵 ∖ (𝐵 ∖ 𝑏)))
42	41	eqeq2d 2632	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏))))
43		eqcom 2629	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
44	42, 43	syl6bb 276	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏))
45		dfss4 3858	. . . . . . . . 9 ⊢ (𝑏 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
46	45	biimpi 206	. . . . . . . 8 ⊢ (𝑏 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
47	46	adantl 482	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
48	39, 44, 47	rspcedvd 3317	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
49	14, 36, 48	syl2an 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
50	49	3ad2antl1 1223	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
51		simp13 1093	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑎 = (𝐵 ∖ 𝑠))
52		fveq2 6191	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝐼‘𝑎) = (𝐼‘(𝐵 ∖ 𝑠)))
53	52	ineq1d 3813	. . . . . . 7 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)))
54		ineq1 3807	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝑎 ∩ 𝑏) = ((𝐵 ∖ 𝑠) ∩ 𝑏))
55	54	fveq2d 6195	. . . . . . 7 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝐼‘(𝑎 ∩ 𝑏)) = (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)))
56	53, 55	sseq12d 3634	. . . . . 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 6191	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → (𝐼‘𝑏) = (𝐼‘(𝐵 ∖ 𝑡)))
59	58	ineq2d 3814	. . . . . . 7 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))))
60		ineq2 3808	. . . . . . . . 9 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐵 ∖ 𝑠) ∩ 𝑏) = ((𝐵 ∖ 𝑠) ∩ (𝐵 ∖ 𝑡)))
61		difundi 3879	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝑠 ∪ 𝑡)) = ((𝐵 ∖ 𝑠) ∩ (𝐵 ∖ 𝑡))
62	60, 61	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐵 ∖ 𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠 ∪ 𝑡)))
63	62	fveq2d 6195	. . . . . . 7 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))))
64	59, 63	sseq12d 3634	. . . . . 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 1084	. . . . 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 1091	. . . . . . . 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 38351	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵))
69	68, 14	jca 554	. . . . . . . 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 7879	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
72	71	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
73		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
74		difssd 3738	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
75	73, 74	sselpwd 4807	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
76	72, 75	ffvelrnd 6360	. . . . . . . . . 10 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
77	76	elpwid 4170	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
78		orc 400	. . . . . . . . 9 ⊢ ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵 ∖ 𝑡)) ⊆ 𝐵))
79		inss 3842	. . . . . . . . 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 3738	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠 ∪ 𝑡)) ⊆ 𝐵)
82	73, 81	sselpwd 4807	. . . . . . . . . 10 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠 ∪ 𝑡)) ∈ 𝒫 𝐵)
83	72, 82	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ∈ 𝒫 𝐵)
84	83	elpwid 4170	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵)
85	80, 84	jca 554	. . . . . . 7 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵) ∧ 𝐵 ∈ V) → (((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵))
86		sscon34b 38317	. . . . . . 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 3881	. . . . . . . 8 ⊢ (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
89	88	sseq2i 3630	. . . . . . 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 1092	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑠 ∈ 𝒫 𝐵)
94		rp-simp2 38087	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑡 ∈ 𝒫 𝐵)
95		simpl2 1065	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
96		simpl3 1066	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵))
97		eqid 2622	. . . . . . . . . 10 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
98		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
99		simprl 794	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
100	99	elpwid 4170	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ⊆ 𝐵)
101		simprr 796	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
102	101	elpwid 4170	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ⊆ 𝐵)
103	100, 102	unssd 3789	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ⊆ 𝐵)
104	98, 103	sselpwd 4807	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝐵)
105	104	3ad2antl2 1224	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝐵)
106		eqid 2622	. . . . . . . . . 10 ⊢ ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡))
107	67, 12, 95, 96, 97, 105, 106	dssmapfv3d 38313	. . . . . . . . 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 1064	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝜑)
109	67, 12, 13	ntrclsfv1 38353	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
110	109	fveq1d 6193	. . . . . . . . . 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 2658	. . . . . . . 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 794	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
114		eqid 2622	. . . . . . . . . . 11 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
115	67, 12, 95, 96, 97, 113, 114	dssmapfv3d 38313	. . . . . . . . . 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 6193	. . . . . . . . . . 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 2658	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) = (𝐾‘𝑠))
119		simprr 796	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_𝑚 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
120		eqid 2622	. . . . . . . . . . 11 ⊢ ((𝐷‘𝐼)‘𝑡) = ((𝐷‘𝐼)‘𝑡)
121	67, 12, 95, 96, 97, 119, 120	dssmapfv3d 38313	. . . . . . . . . 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 6193	. . . . . . . . . . 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 2658	. . . . . . . . 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 3768	. . . . . . . 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 3634	. . . . . . 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 1334	. . . . . 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 294	. . . . 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 294	. . . 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 4884	. . 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 4884	. 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 272	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 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-frege1 38084

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859

This theorem is referenced by: (None)

W3C validator