ntrclsk3 - Mathbox for Richard Penner

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Theorem ntrclsk3 44059

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 6858	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐼‘𝑠) = (𝐼‘𝑎))
2	1	ineq1d 4182	. . . 4 ⊢ (𝑠 = 𝑎 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑡)))
3		ineq1 4176	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝑠 ∩ 𝑡) = (𝑎 ∩ 𝑡))
4	3	fveq2d 6862	. . . 4 ⊢ (𝑠 = 𝑎 → (𝐼‘(𝑠 ∩ 𝑡)) = (𝐼‘(𝑎 ∩ 𝑡)))
5	2, 4	sseq12d 3980	. . 3 ⊢ (𝑠 = 𝑎 → (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ((𝐼‘𝑎) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑎 ∩ 𝑡))))
6		fveq2 6858	. . . . 5 ⊢ (𝑡 = 𝑏 → (𝐼‘𝑡) = (𝐼‘𝑏))
7	6	ineq2d 4183	. . . 4 ⊢ (𝑡 = 𝑏 → ((𝐼‘𝑎) ∩ (𝐼‘𝑡)) = ((𝐼‘𝑎) ∩ (𝐼‘𝑏)))
8		ineq2 4177	. . . . 5 ⊢ (𝑡 = 𝑏 → (𝑎 ∩ 𝑡) = (𝑎 ∩ 𝑏))
9	8	fveq2d 6862	. . . 4 ⊢ (𝑡 = 𝑏 → (𝐼‘(𝑎 ∩ 𝑡)) = (𝐼‘(𝑎 ∩ 𝑏)))
10	7, 9	sseq12d 3980	. . 3 ⊢ (𝑡 = 𝑏 → (((𝐼‘𝑎) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑎 ∩ 𝑡)) ↔ ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) ⊆ (𝐼‘(𝑎 ∩ 𝑏))))
11	5, 10	cbvral2vw 3219	. 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 44023	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
15		difssd 4100	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ⊆ 𝐵)
16	14, 15	sselpwd 5283	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
18		elpwi 4570	. . . 4 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
19		simpl 482	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → 𝐵 ∈ V)
20		difssd 4100	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ⊆ 𝐵)
21	19, 20	sselpwd 5283	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ∈ 𝒫 𝐵)
22		simpr 484	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → 𝑠 = (𝐵 ∖ 𝑎))
23	22	difeq2d 4089	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑎)))
24	23	eqeq2d 2740	. . . . . 6 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎))))
25		eqcom 2736	. . . . . 6 ⊢ (𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
26	24, 25	bitrdi 287	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎))
27		dfss4 4232	. . . . . . 7 ⊢ (𝑎 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
28	27	biimpi 216	. . . . . 6 ⊢ (𝑎 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
29	28	adantl 481	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
30	21, 26, 29	rspcedvd 3590	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
31	14, 18, 30	syl2an 596	. . 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 4100	. . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ⊆ 𝐵)
34	14, 33	sselpwd 5283	. . . . 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 4570	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵)
37		simpl 482	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → 𝐵 ∈ V)
38		difssd 4100	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ⊆ 𝐵)
39	37, 38	sselpwd 5283	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ∈ 𝒫 𝐵)
40		simpr 484	. . . . . . . . . 10 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → 𝑡 = (𝐵 ∖ 𝑏))
41	40	difeq2d 4089	. . . . . . . . 9 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝐵 ∖ 𝑡) = (𝐵 ∖ (𝐵 ∖ 𝑏)))
42	41	eqeq2d 2740	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏))))
43		eqcom 2736	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
44	42, 43	bitrdi 287	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏))
45		dfss4 4232	. . . . . . . . 9 ⊢ (𝑏 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
46	45	biimpi 216	. . . . . . . 8 ⊢ (𝑏 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
47	46	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
48	39, 44, 47	rspcedvd 3590	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
49	14, 36, 48	syl2an 596	. . . . 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 6858	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝐼‘𝑎) = (𝐼‘(𝐵 ∖ 𝑠)))
53	52	ineq1d 4182	. . . . . . 7 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → ((𝐼‘𝑎) ∩ (𝐼‘𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)))
54		ineq1 4176	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝑎 ∩ 𝑏) = ((𝐵 ∖ 𝑠) ∩ 𝑏))
55	54	fveq2d 6862	. . . . . . 7 ⊢ (𝑎 = (𝐵 ∖ 𝑠) → (𝐼‘(𝑎 ∩ 𝑏)) = (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)))
56	53, 55	sseq12d 3980	. . . . . 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 6858	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → (𝐼‘𝑏) = (𝐼‘(𝐵 ∖ 𝑡)))
59	58	ineq2d 4183	. . . . . . 7 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘𝑏)) = ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))))
60		ineq2 4177	. . . . . . . . 9 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐵 ∖ 𝑠) ∩ 𝑏) = ((𝐵 ∖ 𝑠) ∩ (𝐵 ∖ 𝑡)))
61		difundi 4253	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝑠 ∪ 𝑡)) = ((𝐵 ∖ 𝑠) ∩ (𝐵 ∖ 𝑡))
62	60, 61	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → ((𝐵 ∖ 𝑠) ∩ 𝑏) = (𝐵 ∖ (𝑠 ∪ 𝑡)))
63	62	fveq2d 6862	. . . . . . 7 ⊢ (𝑏 = (𝐵 ∖ 𝑡) → (𝐼‘((𝐵 ∖ 𝑠) ∩ 𝑏)) = (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))))
64	59, 63	sseq12d 3980	. . . . . 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 1135	. . . . 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	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝜑)
67		ntrcls.o	. . . . . . . . 9 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
68	67, 12, 13	ntrclsiex 44042	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
69	68, 14	jca 511	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V))
70		elmapi 8822	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
71	70	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
72		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
73		difssd 4100	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
74	72, 73	sselpwd 5283	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
75	71, 74	ffvelcdmd 7057	. . . . . . . . . 10 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
76	75	elpwid 4572	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
77		orc 867	. . . . . . . . 9 ⊢ ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵 ∖ 𝑡)) ⊆ 𝐵))
78		inss 4211	. . . . . . . . 9 ⊢ (((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 ∨ (𝐼‘(𝐵 ∖ 𝑡)) ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵)
79	76, 77, 78	3syl 18	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵)
80		difssd 4100	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠 ∪ 𝑡)) ⊆ 𝐵)
81	72, 80	sselpwd 5283	. . . . . . . . . 10 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐵 ∖ (𝑠 ∪ 𝑡)) ∈ 𝒫 𝐵)
82	71, 81	ffvelcdmd 7057	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ∈ 𝒫 𝐵)
83	82	elpwid 4572	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵)
84	79, 83	jca 511	. . . . . . 7 ⊢ ((𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ∧ 𝐵 ∈ V) → (((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡))) ⊆ 𝐵))
85		sscon34b 4267	. . . . . . 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))))))
86	66, 69, 84, 85	4syl 19	. . . . . 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))))))
87		difindi 4255	. . . . . . . 8 ⊢ (𝐵 ∖ ((𝐼‘(𝐵 ∖ 𝑠)) ∩ (𝐼‘(𝐵 ∖ 𝑡)))) = ((𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ∪ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
88	87	sseq2i 3976	. . . . . . 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)))))
89	88	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))))))
90	66, 14	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝐵 ∈ V)
91	66, 68	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
92		simp12 1205	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑠 ∈ 𝒫 𝐵)
93		rp-simp2 43782	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑡 ∈ 𝒫 𝐵)
94		simpl2 1193	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
95		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
96		eqid 2729	. . . . . . . . . 10 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
97		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝐵 ∈ V)
98		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
99	98	elpwid 4572	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ⊆ 𝐵)
100		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
101	100	elpwid 4572	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ⊆ 𝐵)
102	99, 101	unssd 4155	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ⊆ 𝐵)
103	97, 102	sselpwd 5283	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝐵)
104	103	3ad2antl2 1187	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝐵)
105		eqid 2729	. . . . . . . . . 10 ⊢ ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡))
106	67, 12, 94, 95, 96, 104, 105	dssmapfv3d 44008	. . . . . . . . 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)))))
107		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝜑)
108	67, 12, 13	ntrclsfv1 44044	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
109	108	fveq1d 6860	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐷‘𝐼)‘(𝑠 ∪ 𝑡)) = (𝐾‘(𝑠 ∪ 𝑡)))
110	107, 109	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)))
111	106, 110	eqtr3d 2766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝑠 ∪ 𝑡)))) = (𝐾‘(𝑠 ∪ 𝑡)))
112		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑠 ∈ 𝒫 𝐵)
113		eqid 2729	. . . . . . . . . . 11 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
114	67, 12, 94, 95, 96, 112, 113	dssmapfv3d 44008	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
115	108	fveq1d 6860	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
116	107, 115	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
117	114, 116	eqtr3d 2766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) = (𝐾‘𝑠))
118		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → 𝑡 ∈ 𝒫 𝐵)
119		eqid 2729	. . . . . . . . . . 11 ⊢ ((𝐷‘𝐼)‘𝑡) = ((𝐷‘𝐼)‘𝑡)
120	67, 12, 94, 95, 96, 118, 119	dssmapfv3d 44008	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
121	108	fveq1d 6860	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐷‘𝐼)‘𝑡) = (𝐾‘𝑡))
122	107, 121	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → ((𝐷‘𝐼)‘𝑡) = (𝐾‘𝑡))
123	120, 122	eqtr3d 2766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ V ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) ∧ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 ∈ 𝒫 𝐵)) → (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))) = (𝐾‘𝑡))
124	117, 123	uneq12d 4132	. . . . . . . 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)))
125	111, 124	sseq12d 3980	. . . . . . 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))))
126	66, 90, 91, 92, 93, 125	syl32anc 1380	. . . . . 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))))
127	86, 89, 126	3bitrd 305	. . . . 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))))
128	57, 65, 127	3bitrd 305	. . . 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))))
129	35, 50, 128	ralxfrd2 5367	. . 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))))
130	17, 31, 129	ralxfrd2 5367	. 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))))
131	11, 130	bitrid 283	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 𝒫 cpw 4563 class class class wbr 5107 ↦ cmpt 5188 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-frege1 43779

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-map 8801

This theorem is referenced by: (None)

W3C validator