Theorem ntrclsiso 44056

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 3972	. . . . 5 ⊢ (𝑠 = 𝑏 → (𝑠 ⊆ 𝑡 ↔ 𝑏 ⊆ 𝑡))
2		fveq2 6858	. . . . . 6 ⊢ (𝑠 = 𝑏 → (𝐼‘𝑠) = (𝐼‘𝑏))
3	2	sseq1d 3978	. . . . 5 ⊢ (𝑠 = 𝑏 → ((𝐼‘𝑠) ⊆ (𝐼‘𝑡) ↔ (𝐼‘𝑏) ⊆ (𝐼‘𝑡)))
4	1, 3	imbi12d 344	. . . 4 ⊢ (𝑠 = 𝑏 → ((𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ (𝑏 ⊆ 𝑡 → (𝐼‘𝑏) ⊆ (𝐼‘𝑡))))
5		sseq2 3973	. . . . 5 ⊢ (𝑡 = 𝑎 → (𝑏 ⊆ 𝑡 ↔ 𝑏 ⊆ 𝑎))
6		fveq2 6858	. . . . . 6 ⊢ (𝑡 = 𝑎 → (𝐼‘𝑡) = (𝐼‘𝑎))
7	6	sseq2d 3979	. . . . 5 ⊢ (𝑡 = 𝑎 → ((𝐼‘𝑏) ⊆ (𝐼‘𝑡) ↔ (𝐼‘𝑏) ⊆ (𝐼‘𝑎)))
8	5, 7	imbi12d 344	. . . 4 ⊢ (𝑡 = 𝑎 → ((𝑏 ⊆ 𝑡 → (𝐼‘𝑏) ⊆ (𝐼‘𝑡)) ↔ (𝑏 ⊆ 𝑎 → (𝐼‘𝑏) ⊆ (𝐼‘𝑎))))
9	4, 8	cbvral2vw 3219	. . 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 3265	. . 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 275	. 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 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝜑)
13		ntrcls.d	. . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵)
14		ntrcls.r	. . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾)
15	13, 14	ntrclsbex 44023	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
16	12, 15	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
17		difssd 4100	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
18	16, 17	sselpwd 5283	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
19		elpwi 4570	. . . 4 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
20		simpl 482	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → 𝐵 ∈ V)
21		difssd 4100	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ⊆ 𝐵)
22	20, 21	sselpwd 5283	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ 𝑎) ∈ 𝒫 𝐵)
23		simpr 484	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → 𝑠 = (𝐵 ∖ 𝑎))
24	23	difeq2d 4089	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑎)))
25	24	eqeq2d 2740	. . . . . 6 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ 𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎))))
26		eqcom 2736	. . . . . 6 ⊢ (𝑎 = (𝐵 ∖ (𝐵 ∖ 𝑎)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
27	25, 26	bitrdi 287	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑎)) → (𝑎 = (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎))
28		dfss4 4232	. . . . . . 7 ⊢ (𝑎 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
29	28	biimpi 216	. . . . . 6 ⊢ (𝑎 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
30	29	adantl 481	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑎)) = 𝑎)
31	22, 27, 30	rspcedvd 3590	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝑎 ⊆ 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
32	15, 19, 31	syl2an 596	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑎 = (𝐵 ∖ 𝑠))
33		simpl1 1192	. . . . . 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 4100	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ⊆ 𝐵)
36	34, 35	sselpwd 5283	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
37		elpwi 4570	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵)
38		simpl 482	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → 𝐵 ∈ V)
39		difssd 4100	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ⊆ 𝐵)
40	38, 39	sselpwd 5283	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ 𝑏) ∈ 𝒫 𝐵)
41		simpr 484	. . . . . . . . . 10 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → 𝑡 = (𝐵 ∖ 𝑏))
42	41	difeq2d 4089	. . . . . . . . 9 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝐵 ∖ 𝑡) = (𝐵 ∖ (𝐵 ∖ 𝑏)))
43	42	eqeq2d 2740	. . . . . . . 8 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ 𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏))))
44		eqcom 2736	. . . . . . . 8 ⊢ (𝑏 = (𝐵 ∖ (𝐵 ∖ 𝑏)) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
45	43, 44	bitrdi 287	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) ∧ 𝑡 = (𝐵 ∖ 𝑏)) → (𝑏 = (𝐵 ∖ 𝑡) ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏))
46		dfss4 4232	. . . . . . . . 9 ⊢ (𝑏 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
47	46	biimpi 216	. . . . . . . 8 ⊢ (𝑏 ⊆ 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
48	47	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑏)) = 𝑏)
49	40, 45, 48	rspcedvd 3590	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑏 ⊆ 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
50	15, 37, 49	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
51	50	3ad2antl1 1186	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑏 ∈ 𝒫 𝐵) → ∃𝑡 ∈ 𝒫 𝐵𝑏 = (𝐵 ∖ 𝑡))
52		simp12 1205	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑠 ∈ 𝒫 𝐵)
53	52	elpwid 4572	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑠 ⊆ 𝐵)
54		simp2 1137	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑡 ∈ 𝒫 𝐵)
55	54	elpwid 4572	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑡 ⊆ 𝐵)
56		sscon34b 4267	. . . . . . . 8 ⊢ ((𝑠 ⊆ 𝐵 ∧ 𝑡 ⊆ 𝐵) → (𝑠 ⊆ 𝑡 ↔ (𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠)))
57	53, 55, 56	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝑠 ⊆ 𝑡 ↔ (𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠)))
58	57	bicomd 223	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ 𝑡))
59		simp11 1204	. . . . . . . . . . 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 44042	. . . . . . . . . . 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 8822	. . . . . . . . . 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 4100	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑡) ⊆ 𝐵)
67	65, 66	sselpwd 5283	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
68	64, 67	ffvelcdmd 7057	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘(𝐵 ∖ 𝑡)) ∈ 𝒫 𝐵)
69	68	elpwid 4572	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘(𝐵 ∖ 𝑡)) ⊆ 𝐵)
70		difssd 4100	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
71	65, 70	sselpwd 5283	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
72	64, 71	ffvelcdmd 7057	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
73	72	elpwid 4572	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
74		sscon34b 4267	. . . . . . 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 584	. . . . . 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 344	. . . . 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 1138	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑏 = (𝐵 ∖ 𝑡))
78		simp13 1206	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → 𝑎 = (𝐵 ∖ 𝑠))
79	77, 78	sseq12d 3980	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝑏 ⊆ 𝑎 ↔ (𝐵 ∖ 𝑡) ⊆ (𝐵 ∖ 𝑠)))
80	77	fveq2d 6862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘𝑏) = (𝐼‘(𝐵 ∖ 𝑡)))
81	78	fveq2d 6862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐼‘𝑎) = (𝐼‘(𝐵 ∖ 𝑠)))
82	80, 81	sseq12d 3980	. . . . . 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 344	. . . . 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 44044	. . . . . . . . . 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 6860	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
87		eqid 2729	. . . . . . . . 9 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
88		eqid 2729	. . . . . . . . 9 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
89	60, 13, 65, 62, 87, 52, 88	dssmapfv3d 44008	. . . . . . . 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 2766	. . . . . . 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 44044	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐷‘𝐼) = 𝐾)
93	92	fveq1d 6860	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → ((𝐷‘𝐼)‘𝑡) = (𝐾‘𝑡))
94		eqid 2729	. . . . . . . . 9 ⊢ ((𝐷‘𝐼)‘𝑡) = ((𝐷‘𝐼)‘𝑡)
95	60, 13, 65, 62, 87, 54, 94	dssmapfv3d 44008	. . . . . . . 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 2766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑎 = (𝐵 ∖ 𝑠)) ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑏 = (𝐵 ∖ 𝑡)) → (𝐾‘𝑡) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑡))))
97	90, 96	sseq12d 3980	. . . . . 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 340	. . . . 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 311	. . . 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 5367	. . 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 5367	. 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	bitrid 283	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∖ cdif 3911   ⊆ 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

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)