Theorem ntrclsiso 38365

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that either is isotonic hold equally. (Contributed by RP, 3-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
ntrclsiso	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡))))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑠,𝑡   𝑗,𝐼,𝑘,𝑠,𝑡   𝜑,𝑖,𝑗,𝑘,𝑠,𝑡

Allowed substitution hints:   𝐷(𝑡,𝑖,𝑗,𝑘,𝑠)   𝐼(𝑖)   𝐾(𝑡,𝑖,𝑗,𝑘,𝑠)   𝑂(𝑡,𝑖,𝑗,𝑘,𝑠)

Proof of Theorem ntrclsiso

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3626	. . . . 5 ⊢ (𝑠 = 𝑏 → (𝑠 ⊆ 𝑡 ↔ 𝑏 ⊆ 𝑡))
2		fveq2 6191	. . . . . 6 ⊢ (𝑠 = 𝑏 → (𝐼‘𝑠) = (𝐼‘𝑏))
3	2	sseq1d 3632	. . . . 5 ⊢ (𝑠 = 𝑏 → ((𝐼‘𝑠) ⊆ (𝐼‘𝑡) ↔ (𝐼‘𝑏) ⊆ (𝐼‘𝑡)))
4	1, 3	imbi12d 334	. . . 4 ⊢ (𝑠 = 𝑏 → ((𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ (𝑏 ⊆ 𝑡 → (𝐼‘𝑏) ⊆ (𝐼‘𝑡))))
5		sseq2 3627	. . . . 5 ⊢ (𝑡 = 𝑎 → (𝑏 ⊆ 𝑡 ↔ 𝑏 ⊆ 𝑎))
6		fveq2 6191	. . . . . 6 ⊢ (𝑡 = 𝑎 → (𝐼‘𝑡) = (𝐼‘𝑎))
7	6	sseq2d 3633	. . . . 5 ⊢ (𝑡 = 𝑎 → ((𝐼‘𝑏) ⊆ (𝐼‘𝑡) ↔ (𝐼‘𝑏) ⊆ (𝐼‘𝑎)))
8	5, 7	imbi12d 334	. . . 4 ⊢ (𝑡 = 𝑎 → ((𝑏 ⊆ 𝑡 → (𝐼‘𝑏) ⊆ (𝐼‘𝑡)) ↔ (𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎))))
9	4, 8	cbvral2v 3179	. . 3 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑏 ∈ 𝒫 𝐵∀𝑎 ∈ 𝒫 𝐵(𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎)))
10		ralcom 3098	. . 3 ⊢ (∀𝑏 ∈ 𝒫 𝐵∀𝑎 ∈ 𝒫 𝐵(𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎)) ↔ ∀𝑎 ∈ 𝒫 𝐵∀𝑏 ∈ 𝒫 𝐵(𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎)))
11	9, 10	bitri 264	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑎 ∈ 𝒫 𝐵∀𝑏 ∈ 𝒫 𝐵(𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎)))
12		simpl 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝜑)
13		ntrcls.d	. . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵)
14		ntrcls.r	. . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾)
15	13, 14	ntrclsbex 38332	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
16	12, 15	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
17		difssd 3738	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
18	16, 17	sselpwd 4807	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
19		elpwi 4168	. . . 4 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
20		simpl 473	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → 𝐵 ∈ V)
21		difssd 3738	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ⊆ 𝐵)
22	20, 21	sselpwd 4807	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ∈ 𝒫 𝐵)
23		simpr 477	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → 𝑠 = (𝐵 ∖ 𝑎))
24	23	difeq2d 3728	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑎)))
25	24	eqeq2d 2632	. . . . . 6 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎))))
26		eqcom 2629	. . . . . 6 ⊢ (𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
27	25, 26	syl6bb 276	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎))
28		dfss4 3858	. . . . . . 7 ⊢ (𝑎 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
29	28	biimpi 206	. . . . . 6 ⊢ (𝑎 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
30	29	adantl 482	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
31	22, 27, 30	rspcedvd 3317	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
32	15, 19, 31	syl2an 494	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
33		simpl1 1064	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
34	33, 15	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
35		difssd 3738	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ⊆ 𝐵)
36	34, 35	sselpwd 4807	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
37		elpwi 4168	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵)
38		simpl 473	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → 𝐵 ∈ V)
39		difssd 3738	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ⊆ 𝐵)
40	38, 39	sselpwd 4807	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ∈ 𝒫 𝐵)
41		simpr 477	. . . . . . . . . 10 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → 𝑡 = (𝐵 ∖ 𝑏))
42	41	difeq2d 3728	. . . . . . . . 9 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝐵 ∖ 𝑡) = (𝐵 ∖ (𝐵 ∖ 𝑏)))
43	42	eqeq2d 2632	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏))))
44		eqcom 2629	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
45	43, 44	syl6bb 276	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏))
46		dfss4 3858	. . . . . . . . 9 ⊢ (𝑏 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
47	46	biimpi 206	. . . . . . . 8 ⊢ (𝑏 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
48	47	adantl 482	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
49	40, 45, 48	rspcedvd 3317	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
50	15, 37, 49	syl2an 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
51	50	3ad2antl1 1223	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
52		simp12 1092	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑠 ∈ 𝒫 𝐵)
53	52	elpwid 4170	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑠 ⊆ 𝐵)
54		simp2 1062	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑡 ∈ 𝒫 𝐵)
55	54	elpwid 4170	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑡 ⊆ 𝐵)
56		sscon34b 38317	. . . . . . . 8 ⊢ ((𝑠 ⊆ 𝐵 ∧ 𝑡 ⊆ 𝐵) → (𝑠 ⊆ 𝑡 ↔ (𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠)))
57	53, 55, 56	syl2anc 693	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝑠 ⊆ 𝑡 ↔ (𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠)))
58	57	bicomd 213	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ 𝑡))
59		simp11 1091	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝜑)
60		ntrcls.o	. . . . . . . . . . . 12 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
61	60, 13, 14	ntrclsiex 38351	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
62	59, 61	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
63		elmapi 7879	. . . . . . . . . 10 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64	62, 63	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
65	59, 15	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝐵 ∈ V)
66		difssd 3738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑡) ⊆ 𝐵)
67	65, 66	sselpwd 4807	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
68	64, 67	ffvelrnd 6360	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘(𝐵 ∖ 𝑡)) ∈ 𝒫 𝐵)
69	68	elpwid 4170	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘(𝐵 ∖ 𝑡)) ⊆ 𝐵)
70		difssd 3738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
71	65, 70	sselpwd 4807	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
72	64, 71	ffvelrnd 6360	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
73	72	elpwid 4170	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
74		sscon34b 38317	. . . . . . 7 ⊢ (((𝐼‘(𝐵 ∖ 𝑡)) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ 𝑡)) ⊆ (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡)))))
75	69, 73, 74	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐼‘(𝐵 ∖ 𝑡)) ⊆ (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡)))))
76	58, 75	imbi12d 334	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (((𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠) → (𝐼‘(𝐵 ∖ 𝑡)) ⊆ (𝐼‘(𝐵 ∖ 𝑠))) ↔ (𝑠 ⊆ 𝑡 → (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))))
77		simp3 1063	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑏 = (𝐵 ∖ 𝑡))
78		simp13 1093	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑎 = (𝐵 ∖ 𝑠))
79	77, 78	sseq12d 3634	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝑏 ⊆ 𝑎 ↔ (𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠)))
80	77	fveq2d 6195	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘𝑏) = (𝐼‘(𝐵 ∖ 𝑡)))
81	78	fveq2d 6195	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘𝑎) = (𝐼‘(𝐵 ∖ 𝑠)))
82	80, 81	sseq12d 3634	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐼‘𝑏) ⊆ (𝐼‘𝑎) ↔ (𝐼‘(𝐵 ∖ 𝑡)) ⊆ (𝐼‘(𝐵 ∖ 𝑠))))
83	79, 82	imbi12d 334	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎)) ↔ ((𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠) → (𝐼‘(𝐵 ∖ 𝑡)) ⊆ (𝐼‘(𝐵 ∖ 𝑠)))))
84	60, 13, 14	ntrclsfv1 38353	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
85	59, 84	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐷‘𝐼) = 𝐾)
86	85	fveq1d 6193	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
87		eqid 2622	. . . . . . . . 9 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
88		eqid 2622	. . . . . . . . 9 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
89	60, 13, 65, 62, 87, 52, 88	dssmapfv3d 38313	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
90	86, 89	eqtr3d 2658	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐾‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
91	59, 14	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝐼𝐷𝐾)
92	60, 13, 91	ntrclsfv1 38353	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐷‘𝐼) = 𝐾)
93	92	fveq1d 6193	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐷‘𝐼)‘𝑡) = (𝐾‘𝑡))
94		eqid 2622	. . . . . . . . 9 ⊢ ((𝐷‘𝐼)‘𝑡) = ((𝐷‘𝐼)‘𝑡)
95	60, 13, 65, 62, 87, 54, 94	dssmapfv3d 38313	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐷‘𝐼)‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
96	93, 95	eqtr3d 2658	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐾‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
97	90, 96	sseq12d 3634	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐾‘𝑠) ⊆ (𝐾‘𝑡) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡)))))
98	97	imbi2d 330	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ (𝑠 ⊆ 𝑡 → (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))))
99	76, 83, 98	3bitr4d 300	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎)) ↔ (𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡))))
100	36, 51, 99	ralxfrd2 4884	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) → (∀𝑏 ∈ 𝒫 𝐵(𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎)) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡))))
101	18, 32, 100	ralxfrd2 4884	. 2 ⊢ (𝜑 → (∀𝑎 ∈ 𝒫 𝐵∀𝑏 ∈ 𝒫 𝐵(𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡))))
102	11, 101	syl5bb 272	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ 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

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)