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Theorem cfilucfil 24771

Description: Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 25479. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

**Hypothesis**
Ref	Expression
metust.1	⊢ 𝐹 = ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))

**Assertion**
Ref	Expression
cfilucfil	⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))

Distinct variable groups: 𝐷,𝑎 𝑋,𝑎 𝐹,𝑎,𝑥 𝑥,𝐷,𝑦 𝑥,𝐹,𝑦 𝑥,𝑋,𝑦,𝑎 𝑦,𝐷 𝐶,𝑎,𝑥,𝑦

Proof of Theorem cfilucfil Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metust.1	. . . . 5 ⊢ 𝐹 = ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))
2	1	metust 24770	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
3		cfilufbas 24500	. . . 4 ⊢ ((((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) → 𝐶 ∈ (fBas‘𝑋))
4	2, 3	sylan 592	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) → 𝐶 ∈ (fBas‘𝑋))
5		simpllr 788	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ (PsMet‘𝑋))
6		psmetf 24518	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
7		ffun 6712	. . . . . 6 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → Fun 𝐷)
8	5, 6, 7	3syl 19	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → Fun 𝐷)
9	2	ad2antrr 739	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
10		simplr 781	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)))
11	1	metustfbas 24769	. . . . . . . 8 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
12	11	ad2antrr 739	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
13		cnvimass 6086	. . . . . . . 8 ⊢ (^◡𝐷 “ (0[,)𝑥)) ⊆ dom 𝐷
14		fdm 6719	. . . . . . . . 9 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → dom 𝐷 = (𝑋 × 𝑋))
15	5, 6, 14	3syl 19	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → dom 𝐷 = (𝑋 × 𝑋))
16	13, 15	sseqtrid 3980	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)𝑥)) ⊆ (𝑋 × 𝑋))
17		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
18	17	rphalfcld 13092	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / 2) ∈ ℝ⁺)
19		eqidd 2766	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)(𝑥 / 2))) = (^◡𝐷 “ (0[,)(𝑥 / 2))))
20		oveq2 7427	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑥 / 2) → (0[,)𝑎) = (0[,)(𝑥 / 2)))
21	20	imaeq2d 6064	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑥 / 2) → (^◡𝐷 “ (0[,)𝑎)) = (^◡𝐷 “ (0[,)(𝑥 / 2))))
22	21	rspceeqv 3606	. . . . . . . . . 10 ⊢ (((𝑥 / 2) ∈ ℝ⁺ ∧ (^◡𝐷 “ (0[,)(𝑥 / 2))) = (^◡𝐷 “ (0[,)(𝑥 / 2)))) → ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)(𝑥 / 2))) = (^◡𝐷 “ (0[,)𝑎)))
23	18, 19, 22	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)(𝑥 / 2))) = (^◡𝐷 “ (0[,)𝑎)))
24	1	metustel 24762	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((^◡𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)(𝑥 / 2))) = (^◡𝐷 “ (0[,)𝑎))))
25	24	biimpar 483	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)(𝑥 / 2))) = (^◡𝐷 “ (0[,)𝑎))) → (^◡𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹)
26	5, 23, 25	syl2anc 596	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹)
27		0xr 11275	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ∈ ℝ^*)
29		rpxr 13046	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ^*)
30		0le0 12361	. . . . . . . . . . 11 ⊢ 0 ≤ 0
31	30	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 0)
32		rpre 13045	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
33	32	rehalfcld 12510	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) ∈ ℝ)
34		rphalflt 13067	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) < 𝑥)
35	33, 32, 34	ltled 11377	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) ≤ 𝑥)
36		icossico 13463	. . . . . . . . . 10 ⊢ (((0 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) ∧ (0 ≤ 0 ∧ (𝑥 / 2) ≤ 𝑥)) → (0[,)(𝑥 / 2)) ⊆ (0[,)𝑥))
37	28, 29, 31, 35, 36	syl22anc 852	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (0[,)(𝑥 / 2)) ⊆ (0[,)𝑥))
38		imass2 6106	. . . . . . . . 9 ⊢ ((0[,)(𝑥 / 2)) ⊆ (0[,)𝑥) → (^◡𝐷 “ (0[,)(𝑥 / 2))) ⊆ (^◡𝐷 “ (0[,)𝑥)))
39	17, 37, 38	3syl 19	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)(𝑥 / 2))) ⊆ (^◡𝐷 “ (0[,)𝑥)))
40		sseq1 3963	. . . . . . . . 9 ⊢ (𝑤 = (^◡𝐷 “ (0[,)(𝑥 / 2))) → (𝑤 ⊆ (^◡𝐷 “ (0[,)𝑥)) ↔ (^◡𝐷 “ (0[,)(𝑥 / 2))) ⊆ (^◡𝐷 “ (0[,)𝑥))))
41	40	rspcev 3583	. . . . . . . 8 ⊢ (((^◡𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹 ∧ (^◡𝐷 “ (0[,)(𝑥 / 2))) ⊆ (^◡𝐷 “ (0[,)𝑥))) → ∃𝑤 ∈ 𝐹 𝑤 ⊆ (^◡𝐷 “ (0[,)𝑥)))
42	26, 39, 41	syl2anc 596	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑤 ∈ 𝐹 𝑤 ⊆ (^◡𝐷 “ (0[,)𝑥)))
43		elfg 24083	. . . . . . . 8 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → ((^◡𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ ((^◡𝐷 “ (0[,)𝑥)) ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ (^◡𝐷 “ (0[,)𝑥)))))
44	43	biimpar 483	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ ((^◡𝐷 “ (0[,)𝑥)) ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ (^◡𝐷 “ (0[,)𝑥)))) → (^◡𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹))
45	12, 16, 42, 44	syl12anc 850	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹))
46		cfiluexsm 24501	. . . . . 6 ⊢ ((((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)) ∧ (^◡𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑥)))
47	9, 10, 45, 46	syl3anc 1398	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑥)))
48		funimass2 6623	. . . . . . 7 ⊢ ((Fun 𝐷 ∧ (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑥))) → (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
49	48	ex 418	. . . . . 6 ⊢ (Fun 𝐷 → ((𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑥)) → (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
50	49	reximdv 3182	. . . . 5 ⊢ (Fun 𝐷 → (∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑥)) → ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
51	8, 47, 50	sylc 66	. . . 4 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
52	51	ralrimiva 3159	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
53	4, 52	jca 521	. 2 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) → (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
54		simprl 783	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → 𝐶 ∈ (fBas‘𝑋))
55		oveq2 7427	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (0[,)𝑥) = (0[,)𝑎))
56	55	sseq2d 3970	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎)))
57	56	rexbidv 3191	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎)))
58		simp-4r 796	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
59	58	simprd 501	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
60		simplr 781	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → 𝑎 ∈ ℝ⁺)
61	57, 59, 60	rspcdva 3584	. . . . . . 7 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎))
62		nfv 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋))
63		nfv 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝐶 ∈ (fBas‘𝑋)
64		nfcv 2927	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦ℝ⁺
65		nfre1 3292	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
66	64, 65	nfralw 3314	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
67	63, 66	nfan 1932	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
68	62, 67	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑦((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
69		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)
70	68, 69	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑦(((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
71		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑎 ∈ ℝ⁺
72	70, 71	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑦((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺)
73		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑦(^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣
74	72, 73	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑦(((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
75	54	ad4antr 745	. . . . . . . . . . . 12 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦 ∈ 𝐶) → 𝐶 ∈ (fBas‘𝑋))
76		fbelss 24045	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦 ⊆ 𝑋)
77	75, 76	sylancom 600	. . . . . . . . . . 11 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦 ∈ 𝐶) → 𝑦 ⊆ 𝑋)
78		xpss12 5678	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑋) → (𝑦 × 𝑦) ⊆ (𝑋 × 𝑋))
79	77, 77, 78	syl2anc 596	. . . . . . . . . 10 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦 ∈ 𝐶) → (𝑦 × 𝑦) ⊆ (𝑋 × 𝑋))
80		simp-6r 800	. . . . . . . . . . 11 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦 ∈ 𝐶) → 𝐷 ∈ (PsMet‘𝑋))
81	80, 6, 14	3syl 19	. . . . . . . . . 10 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦 ∈ 𝐶) → dom 𝐷 = (𝑋 × 𝑋))
82	79, 81	sseqtrrd 3975	. . . . . . . . 9 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦 ∈ 𝐶) → (𝑦 × 𝑦) ⊆ dom 𝐷)
83	82	ex 418	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → (𝑦 ∈ 𝐶 → (𝑦 × 𝑦) ⊆ dom 𝐷))
84	74, 83	ralrimi 3265	. . . . . . 7 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∀𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ dom 𝐷)
85		r19.29r 3131	. . . . . . . 8 ⊢ ((∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ ∀𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦 ∈ 𝐶 ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷))
86		sseqin2 4176	. . . . . . . . . . . 12 ⊢ ((𝑦 × 𝑦) ⊆ dom 𝐷 ↔ (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
87	86	bilani 510	. . . . . . . . . . 11 ⊢ (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
88		dminss 6152	. . . . . . . . . . 11 ⊢ (dom 𝐷 ∩ (𝑦 × 𝑦)) ⊆ (^◡𝐷 “ (𝐷 “ (𝑦 × 𝑦)))
89	87, 88	eqsstrrdi 3983	. . . . . . . . . 10 ⊢ (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (𝐷 “ (𝑦 × 𝑦))))
90		imass2 6106	. . . . . . . . . . 11 ⊢ ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) → (^◡𝐷 “ (𝐷 “ (𝑦 × 𝑦))) ⊆ (^◡𝐷 “ (0[,)𝑎)))
91	90	adantr 486	. . . . . . . . . 10 ⊢ (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (^◡𝐷 “ (𝐷 “ (𝑦 × 𝑦))) ⊆ (^◡𝐷 “ (0[,)𝑎)))
92	89, 91	sstrd 3948	. . . . . . . . 9 ⊢ (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)))
93	92	reximi 3105	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐶 ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)))
94	85, 93	syl 18	. . . . . . 7 ⊢ ((∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ ∀𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)))
95	61, 84, 94	syl2anc 596	. . . . . 6 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)))
96		r19.41v 3197	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐶 ((𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ↔ (∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
97		sstr 3946	. . . . . . . 8 ⊢ (((𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → (𝑦 × 𝑦) ⊆ 𝑣)
98	97	reximi 3105	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐶 ((𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
99	96, 98	sylbir 238	. . . . . 6 ⊢ ((∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
100	95, 99	sylancom 600	. . . . 5 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ⁺) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
101		simp-5r 798	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝐹) ∧ 𝑤 ⊆ 𝑣) → 𝐷 ∈ (PsMet‘𝑋))
102		simplr 781	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝐹) ∧ 𝑤 ⊆ 𝑣) → 𝑤 ∈ 𝐹)
103	1	metustel 24762	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑤 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎))))
104	103	biimpa 482	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑤 ∈ 𝐹) → ∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎)))
105	101, 102, 104	syl2anc 596	. . . . . . 7 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝐹) ∧ 𝑤 ⊆ 𝑣) → ∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎)))
106		r19.41v 3197	. . . . . . . 8 ⊢ (∃𝑎 ∈ ℝ⁺ (𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣) ↔ (∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣))
107		sseq1 3963	. . . . . . . . . 10 ⊢ (𝑤 = (^◡𝐷 “ (0[,)𝑎)) → (𝑤 ⊆ 𝑣 ↔ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
108	107	biimpa 482	. . . . . . . . 9 ⊢ ((𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣) → (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
109	108	reximi 3105	. . . . . . . 8 ⊢ (∃𝑎 ∈ ℝ⁺ (𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣) → ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
110	106, 109	sylbir 238	. . . . . . 7 ⊢ ((∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣) → ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
111	105, 110	sylancom 600	. . . . . 6 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝐹) ∧ 𝑤 ⊆ 𝑣) → ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
112	11	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
113		elfg 24083	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣)))
114	113	biimpa 482	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣))
115	112, 114	sylancom 600	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣))
116	115	simprd 501	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣)
117	111, 116	r19.29a 3175	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
118	100, 117	r19.29a 3175	. . . 4 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
119	118	ralrimiva 3159	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
120	2	adantr 486	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
121		iscfilu 24499	. . . 4 ⊢ (((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) → (𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)))
122	120, 121	syl 18	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → (𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)))
123	54, 119, 122	mpbir2and 726	. 2 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)))
124	53, 123	impbida 813	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 class class class wbr 5111 ↦ cmpt 5194 × cxp 5661 ^◡ccnv 5662 dom cdm 5663 ran crn 5664 “ cima 5666 Fun wfun 6534 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 0cc0 11119 ℝ^*cxr 11261 ≤ cle 11263 / cdiv 11890 2c2 12314 ℝ⁺crp 13036 [,)cico 13394 PsMetcpsmet 21560 fBascfbas 21564 filGencfg 21565 UnifOncust 24412 CauFil_uccfilu 24497

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-div 11891 df-nn 12253 df-2 12322 df-rp 13037 df-xneg 13157 df-xadd 13158 df-xmul 13159 df-ico 13398 df-psmet 21568 df-fbas 21573 df-fg 21574 df-fil 24058 df-ust 24413 df-cfilu 24498

This theorem is used by: cfilucfil2 24773

W3C validator