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Theorem utop2nei 22861

Description: For any symmetrical entourage 𝑉 and any relation 𝑀, build a neighborhood of 𝑀. First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)

**Hypothesis**
Ref	Expression
utoptop.1	⊢ 𝐽 = (unifTop‘𝑈)

**Assertion**
Ref	Expression
utop2nei	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘𝑀))

Proof of Theorem utop2nei Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		utoptop.1	. . . . . . . 8 ⊢ 𝐽 = (unifTop‘𝑈)
2		utoptop 22845	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
3	1, 2	eqeltrid 2919	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
4		txtop 22179	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×_t 𝐽) ∈ Top)
5	3, 3, 4	syl2anc 586	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×_t 𝐽) ∈ Top)
6	5	3ad2ant1 1129	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝐽 ×_t 𝐽) ∈ Top)
7	6	adantr 483	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝐽 ×_t 𝐽) ∈ Top)
8		0nei 21738	. . . 4 ⊢ ((𝐽 ×_t 𝐽) ∈ Top → ∅ ∈ ((nei‘(𝐽 ×_t 𝐽))‘∅))
9	7, 8	syl 17	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → ∅ ∈ ((nei‘(𝐽 ×_t 𝐽))‘∅))
10		coeq1 5730	. . . . . . 7 ⊢ (𝑀 = ∅ → (𝑀 ∘ 𝑉) = (∅ ∘ 𝑉))
11		co01 6116	. . . . . . 7 ⊢ (∅ ∘ 𝑉) = ∅
12	10, 11	syl6eq 2874	. . . . . 6 ⊢ (𝑀 = ∅ → (𝑀 ∘ 𝑉) = ∅)
13	12	coeq2d 5735	. . . . 5 ⊢ (𝑀 = ∅ → (𝑉 ∘ (𝑀 ∘ 𝑉)) = (𝑉 ∘ ∅))
14		co02 6115	. . . . 5 ⊢ (𝑉 ∘ ∅) = ∅
15	13, 14	syl6eq 2874	. . . 4 ⊢ (𝑀 = ∅ → (𝑉 ∘ (𝑀 ∘ 𝑉)) = ∅)
16	15	adantl 484	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝑉 ∘ (𝑀 ∘ 𝑉)) = ∅)
17		simpr 487	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → 𝑀 = ∅)
18	17	fveq2d 6676	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → ((nei‘(𝐽 ×_t 𝐽))‘𝑀) = ((nei‘(𝐽 ×_t 𝐽))‘∅))
19	9, 16, 18	3eltr4d 2930	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘𝑀))
20	6	adantr 483	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → (𝐽 ×_t 𝐽) ∈ Top)
21		simpl1 1187	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → 𝑈 ∈ (UnifOn‘𝑋))
22	21, 3	syl 17	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → 𝐽 ∈ Top)
23		simpl2l 1222	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → 𝑉 ∈ 𝑈)
24		simp3 1134	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑀 ⊆ (𝑋 × 𝑋))
25	24	sselda 3969	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → 𝑟 ∈ (𝑋 × 𝑋))
26		xp1st 7723	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (𝑋 × 𝑋) → (1^st ‘𝑟) ∈ 𝑋)
27	25, 26	syl 17	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → (1^st ‘𝑟) ∈ 𝑋)
28	1	utopsnnei 22860	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ (1^st ‘𝑟) ∈ 𝑋) → (𝑉 “ {(1^st ‘𝑟)}) ∈ ((nei‘𝐽)‘{(1^st ‘𝑟)}))
29	21, 23, 27, 28	syl3anc 1367	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → (𝑉 “ {(1^st ‘𝑟)}) ∈ ((nei‘𝐽)‘{(1^st ‘𝑟)}))
30		xp2nd 7724	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (𝑋 × 𝑋) → (2^nd ‘𝑟) ∈ 𝑋)
31	25, 30	syl 17	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → (2^nd ‘𝑟) ∈ 𝑋)
32	1	utopsnnei 22860	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ (2^nd ‘𝑟) ∈ 𝑋) → (𝑉 “ {(2^nd ‘𝑟)}) ∈ ((nei‘𝐽)‘{(2^nd ‘𝑟)}))
33	21, 23, 31, 32	syl3anc 1367	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → (𝑉 “ {(2^nd ‘𝑟)}) ∈ ((nei‘𝐽)‘{(2^nd ‘𝑟)}))
34		eqid 2823	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
35	34, 34	neitx 22217	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐽 ∈ Top) ∧ ((𝑉 “ {(1^st ‘𝑟)}) ∈ ((nei‘𝐽)‘{(1^st ‘𝑟)}) ∧ (𝑉 “ {(2^nd ‘𝑟)}) ∈ ((nei‘𝐽)‘{(2^nd ‘𝑟)}))) → ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘({(1^st ‘𝑟)} × {(2^nd ‘𝑟)})))
36	22, 22, 29, 33, 35	syl22anc 836	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘({(1^st ‘𝑟)} × {(2^nd ‘𝑟)})))
37		fvex 6685	. . . . . . . . . 10 ⊢ (1^st ‘𝑟) ∈ V
38		fvex 6685	. . . . . . . . . 10 ⊢ (2^nd ‘𝑟) ∈ V
39	37, 38	xpsn 6905	. . . . . . . . 9 ⊢ ({(1^st ‘𝑟)} × {(2^nd ‘𝑟)}) = {⟨(1^st ‘𝑟), (2^nd ‘𝑟)⟩}
40	39	fveq2i 6675	. . . . . . . 8 ⊢ ((nei‘(𝐽 ×_t 𝐽))‘({(1^st ‘𝑟)} × {(2^nd ‘𝑟)})) = ((nei‘(𝐽 ×_t 𝐽))‘{⟨(1^st ‘𝑟), (2^nd ‘𝑟)⟩})
41	36, 40	eleqtrdi 2925	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{⟨(1^st ‘𝑟), (2^nd ‘𝑟)⟩}))
42	24	adantr 483	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → 𝑀 ⊆ (𝑋 × 𝑋))
43		xpss 5573	. . . . . . . . . . . . 13 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
44		sstr 3977	. . . . . . . . . . . . 13 ⊢ ((𝑀 ⊆ (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ⊆ (V × V)) → 𝑀 ⊆ (V × V))
45	43, 44	mpan2 689	. . . . . . . . . . . 12 ⊢ (𝑀 ⊆ (𝑋 × 𝑋) → 𝑀 ⊆ (V × V))
46		df-rel 5564	. . . . . . . . . . . 12 ⊢ (Rel 𝑀 ↔ 𝑀 ⊆ (V × V))
47	45, 46	sylibr 236	. . . . . . . . . . 11 ⊢ (𝑀 ⊆ (𝑋 × 𝑋) → Rel 𝑀)
48	42, 47	syl 17	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → Rel 𝑀)
49		1st2nd 7740	. . . . . . . . . 10 ⊢ ((Rel 𝑀 ∧ 𝑟 ∈ 𝑀) → 𝑟 = ⟨(1^st ‘𝑟), (2^nd ‘𝑟)⟩)
50	48, 49	sylancom 590	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → 𝑟 = ⟨(1^st ‘𝑟), (2^nd ‘𝑟)⟩)
51	50	sneqd 4581	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → {𝑟} = {⟨(1^st ‘𝑟), (2^nd ‘𝑟)⟩})
52	51	fveq2d 6676	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → ((nei‘(𝐽 ×_t 𝐽))‘{𝑟}) = ((nei‘(𝐽 ×_t 𝐽))‘{⟨(1^st ‘𝑟), (2^nd ‘𝑟)⟩}))
53	41, 52	eleqtrrd 2918	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑟}))
54		relxp 5575	. . . . . . . . . . 11 ⊢ Rel ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))
55	54	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → Rel ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})))
56		1st2nd 7740	. . . . . . . . . 10 ⊢ ((Rel ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
57	55, 56	sylancom 590	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
58		simpll2 1209	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉))
59	58	simprd 498	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → ^◡𝑉 = 𝑉)
60		simpll1 1208	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → 𝑈 ∈ (UnifOn‘𝑋))
61	58	simpld 497	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → 𝑉 ∈ 𝑈)
62		ustrel 22822	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)
63	60, 61, 62	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → Rel 𝑉)
64		xp1st 7723	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) → (1^st ‘𝑧) ∈ (𝑉 “ {(1^st ‘𝑟)}))
65	64	adantl 484	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → (1^st ‘𝑧) ∈ (𝑉 “ {(1^st ‘𝑟)}))
66		elrelimasn 5955	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑉 → ((1^st ‘𝑧) ∈ (𝑉 “ {(1^st ‘𝑟)}) ↔ (1^st ‘𝑟)𝑉(1^st ‘𝑧)))
67	66	biimpa 479	. . . . . . . . . . . . 13 ⊢ ((Rel 𝑉 ∧ (1^st ‘𝑧) ∈ (𝑉 “ {(1^st ‘𝑟)})) → (1^st ‘𝑟)𝑉(1^st ‘𝑧))
68	63, 65, 67	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → (1^st ‘𝑟)𝑉(1^st ‘𝑧))
69		fvex 6685	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘𝑧) ∈ V
70	37, 69	brcnv 5755	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑟)^◡𝑉(1^st ‘𝑧) ↔ (1^st ‘𝑧)𝑉(1^st ‘𝑟))
71		breq 5070	. . . . . . . . . . . . . 14 ⊢ (^◡𝑉 = 𝑉 → ((1^st ‘𝑟)^◡𝑉(1^st ‘𝑧) ↔ (1^st ‘𝑟)𝑉(1^st ‘𝑧)))
72	70, 71	syl5bbr 287	. . . . . . . . . . . . 13 ⊢ (^◡𝑉 = 𝑉 → ((1^st ‘𝑧)𝑉(1^st ‘𝑟) ↔ (1^st ‘𝑟)𝑉(1^st ‘𝑧)))
73	72	biimpar 480	. . . . . . . . . . . 12 ⊢ ((^◡𝑉 = 𝑉 ∧ (1^st ‘𝑟)𝑉(1^st ‘𝑧)) → (1^st ‘𝑧)𝑉(1^st ‘𝑟))
74	59, 68, 73	syl2anc 586	. . . . . . . . . . 11 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → (1^st ‘𝑧)𝑉(1^st ‘𝑟))
75		simpll3 1210	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → 𝑀 ⊆ (𝑋 × 𝑋))
76		simplr 767	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → 𝑟 ∈ 𝑀)
77		1st2ndbr 7743	. . . . . . . . . . . . 13 ⊢ ((Rel 𝑀 ∧ 𝑟 ∈ 𝑀) → (1^st ‘𝑟)𝑀(2^nd ‘𝑟))
78	47, 77	sylan 582	. . . . . . . . . . . 12 ⊢ ((𝑀 ⊆ (𝑋 × 𝑋) ∧ 𝑟 ∈ 𝑀) → (1^st ‘𝑟)𝑀(2^nd ‘𝑟))
79	75, 76, 78	syl2anc 586	. . . . . . . . . . 11 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → (1^st ‘𝑟)𝑀(2^nd ‘𝑟))
80		xp2nd 7724	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) → (2^nd ‘𝑧) ∈ (𝑉 “ {(2^nd ‘𝑟)}))
81	80	adantl 484	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → (2^nd ‘𝑧) ∈ (𝑉 “ {(2^nd ‘𝑟)}))
82		elrelimasn 5955	. . . . . . . . . . . . 13 ⊢ (Rel 𝑉 → ((2^nd ‘𝑧) ∈ (𝑉 “ {(2^nd ‘𝑟)}) ↔ (2^nd ‘𝑟)𝑉(2^nd ‘𝑧)))
83	82	biimpa 479	. . . . . . . . . . . 12 ⊢ ((Rel 𝑉 ∧ (2^nd ‘𝑧) ∈ (𝑉 “ {(2^nd ‘𝑟)})) → (2^nd ‘𝑟)𝑉(2^nd ‘𝑧))
84	63, 81, 83	syl2anc 586	. . . . . . . . . . 11 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → (2^nd ‘𝑟)𝑉(2^nd ‘𝑧))
85	69, 38, 37	3pm3.2i 1335	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑧) ∈ V ∧ (2^nd ‘𝑟) ∈ V ∧ (1^st ‘𝑟) ∈ V)
86		brcogw 5741	. . . . . . . . . . . . 13 ⊢ ((((1^st ‘𝑧) ∈ V ∧ (2^nd ‘𝑟) ∈ V ∧ (1^st ‘𝑟) ∈ V) ∧ ((1^st ‘𝑧)𝑉(1^st ‘𝑟) ∧ (1^st ‘𝑟)𝑀(2^nd ‘𝑟))) → (1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟))
87	85, 86	mpan 688	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑧)𝑉(1^st ‘𝑟) ∧ (1^st ‘𝑟)𝑀(2^nd ‘𝑟)) → (1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟))
88		fvex 6685	. . . . . . . . . . . . . 14 ⊢ (2^nd ‘𝑧) ∈ V
89	69, 88, 38	3pm3.2i 1335	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑧) ∈ V ∧ (2^nd ‘𝑧) ∈ V ∧ (2^nd ‘𝑟) ∈ V)
90		brcogw 5741	. . . . . . . . . . . . 13 ⊢ ((((1^st ‘𝑧) ∈ V ∧ (2^nd ‘𝑧) ∈ V ∧ (2^nd ‘𝑟) ∈ V) ∧ ((1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟) ∧ (2^nd ‘𝑟)𝑉(2^nd ‘𝑧))) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
91	89, 90	mpan 688	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟) ∧ (2^nd ‘𝑟)𝑉(2^nd ‘𝑧)) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
92	87, 91	sylan 582	. . . . . . . . . . 11 ⊢ ((((1^st ‘𝑧)𝑉(1^st ‘𝑟) ∧ (1^st ‘𝑟)𝑀(2^nd ‘𝑟)) ∧ (2^nd ‘𝑟)𝑉(2^nd ‘𝑧)) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
93	74, 79, 84, 92	syl21anc 835	. . . . . . . . . 10 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
94		df-br 5069	. . . . . . . . . 10 ⊢ ((1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧) ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
95	93, 94	sylib 220	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
96	57, 95	eqeltrd 2915	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)}))) → 𝑧 ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
97	96	ex 415	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → (𝑧 ∈ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) → 𝑧 ∈ (𝑉 ∘ (𝑀 ∘ 𝑉))))
98	97	ssrdv 3975	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) ⊆ (𝑉 ∘ (𝑀 ∘ 𝑉)))
99		simp1 1132	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑈 ∈ (UnifOn‘𝑋))
100		simp2l 1195	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉 ∈ 𝑈)
101		ustssxp 22815	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
102	99, 100, 101	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉 ⊆ (𝑋 × 𝑋))
103		coss1 5728	. . . . . . . . . 10 ⊢ (𝑉 ⊆ (𝑋 × 𝑋) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ⊆ ((𝑋 × 𝑋) ∘ (𝑀 ∘ 𝑉)))
104	102, 103	syl 17	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ⊆ ((𝑋 × 𝑋) ∘ (𝑀 ∘ 𝑉)))
105		coss1 5728	. . . . . . . . . . . 12 ⊢ (𝑀 ⊆ (𝑋 × 𝑋) → (𝑀 ∘ 𝑉) ⊆ ((𝑋 × 𝑋) ∘ 𝑉))
106	24, 105	syl 17	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑀 ∘ 𝑉) ⊆ ((𝑋 × 𝑋) ∘ 𝑉))
107		coss2 5729	. . . . . . . . . . . . 13 ⊢ (𝑉 ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ 𝑉) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
108		xpcoid 6143	. . . . . . . . . . . . 13 ⊢ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)) = (𝑋 × 𝑋)
109	107, 108	sseqtrdi 4019	. . . . . . . . . . . 12 ⊢ (𝑉 ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ 𝑉) ⊆ (𝑋 × 𝑋))
110	102, 109	syl 17	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → ((𝑋 × 𝑋) ∘ 𝑉) ⊆ (𝑋 × 𝑋))
111	106, 110	sstrd 3979	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑀 ∘ 𝑉) ⊆ (𝑋 × 𝑋))
112		coss2 5729	. . . . . . . . . . 11 ⊢ ((𝑀 ∘ 𝑉) ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ (𝑀 ∘ 𝑉)) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
113	112, 108	sseqtrdi 4019	. . . . . . . . . 10 ⊢ ((𝑀 ∘ 𝑉) ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ (𝑀 ∘ 𝑉)) ⊆ (𝑋 × 𝑋))
114	111, 113	syl 17	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → ((𝑋 × 𝑋) ∘ (𝑀 ∘ 𝑉)) ⊆ (𝑋 × 𝑋))
115	104, 114	sstrd 3979	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ⊆ (𝑋 × 𝑋))
116		utopbas 22846	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
117	1	unieqi 4853	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ (unifTop‘𝑈)
118	116, 117	syl6eqr 2876	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ 𝐽)
119	118	sqxpeqd 5589	. . . . . . . . . 10 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (∪ 𝐽 × ∪ 𝐽))
120	34, 34	txuni 22202	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (∪ 𝐽 × ∪ 𝐽) = ∪ (𝐽 ×_t 𝐽))
121	3, 3, 120	syl2anc 586	. . . . . . . . . 10 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (∪ 𝐽 × ∪ 𝐽) = ∪ (𝐽 ×_t 𝐽))
122	119, 121	eqtrd 2858	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ∪ (𝐽 ×_t 𝐽))
123	122	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑋 × 𝑋) = ∪ (𝐽 ×_t 𝐽))
124	115, 123	sseqtrd 4009	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ⊆ ∪ (𝐽 ×_t 𝐽))
125	124	adantr 483	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ⊆ ∪ (𝐽 ×_t 𝐽))
126		eqid 2823	. . . . . . 7 ⊢ ∪ (𝐽 ×_t 𝐽) = ∪ (𝐽 ×_t 𝐽)
127	126	ssnei2 21726	. . . . . 6 ⊢ ((((𝐽 ×_t 𝐽) ∈ Top ∧ ((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑟})) ∧ (((𝑉 “ {(1^st ‘𝑟)}) × (𝑉 “ {(2^nd ‘𝑟)})) ⊆ (𝑉 ∘ (𝑀 ∘ 𝑉)) ∧ (𝑉 ∘ (𝑀 ∘ 𝑉)) ⊆ ∪ (𝐽 ×_t 𝐽))) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑟}))
128	20, 53, 98, 125, 127	syl22anc 836	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟 ∈ 𝑀) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑟}))
129	128	ralrimiva 3184	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → ∀𝑟 ∈ 𝑀 (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑟}))
130	129	adantr 483	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → ∀𝑟 ∈ 𝑀 (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑟}))
131	6	adantr 483	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → (𝐽 ×_t 𝐽) ∈ Top)
132	24, 123	sseqtrd 4009	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑀 ⊆ ∪ (𝐽 ×_t 𝐽))
133	132	adantr 483	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → 𝑀 ⊆ ∪ (𝐽 ×_t 𝐽))
134		simpr 487	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → 𝑀 ≠ ∅)
135	126	neips 21723	. . . 4 ⊢ (((𝐽 ×_t 𝐽) ∈ Top ∧ 𝑀 ⊆ ∪ (𝐽 ×_t 𝐽) ∧ 𝑀 ≠ ∅) → ((𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘𝑀) ↔ ∀𝑟 ∈ 𝑀 (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑟})))
136	131, 133, 134, 135	syl3anc 1367	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → ((𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘𝑀) ↔ ∀𝑟 ∈ 𝑀 (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑟})))
137	130, 136	mpbird 259	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘𝑀))
138	19, 137	pm2.61dane 3106	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×_t 𝐽))‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 {csn 4569 ⟨cop 4575 ∪ cuni 4840 class class class wbr 5068 × cxp 5555 ^◡ccnv 5556 “ cima 5560 ∘ ccom 5561 Rel wrel 5562 ‘cfv 6357 (class class class)co 7158 1^st c1st 7689 2^nd c2nd 7690 Topctop 21503 neicnei 21707 ×_t ctx 22170 UnifOncust 22810 unifTopcutop 22841

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-fin 8515 df-fi 8877 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 df-nei 21708 df-tx 22172 df-ust 22811 df-utop 22842

This theorem is referenced by: (None)

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