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Theorem cfilucfil 23166

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 23869. (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 23165	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
3		cfilufbas 22895	. . . 4 ⊢ ((((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) → 𝐶 ∈ (fBas‘𝑋))
4	2, 3	sylan 583	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) → 𝐶 ∈ (fBas‘𝑋))
5		simpllr 775	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ (PsMet‘𝑋))
6		psmetf 22913	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
7		ffun 6490	. . . . . 6 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → Fun 𝐷)
8	5, 6, 7	3syl 18	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → Fun 𝐷)
9	2	ad2antrr 725	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
10		simplr 768	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)))
11	1	metustfbas 23164	. . . . . . . 8 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
12	11	ad2antrr 725	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
13		cnvimass 5916	. . . . . . . 8 ⊢ (^◡𝐷 “ (0[,)𝑥)) ⊆ dom 𝐷
14		fdm 6495	. . . . . . . . 9 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → dom 𝐷 = (𝑋 × 𝑋))
15	5, 6, 14	3syl 18	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → dom 𝐷 = (𝑋 × 𝑋))
16	13, 15	sseqtrid 3967	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)𝑥)) ⊆ (𝑋 × 𝑋))
17		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
18	17	rphalfcld 12431	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / 2) ∈ ℝ⁺)
19		eqidd 2799	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)(𝑥 / 2))) = (^◡𝐷 “ (0[,)(𝑥 / 2))))
20		oveq2 7143	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑥 / 2) → (0[,)𝑎) = (0[,)(𝑥 / 2)))
21	20	imaeq2d 5896	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑥 / 2) → (^◡𝐷 “ (0[,)𝑎)) = (^◡𝐷 “ (0[,)(𝑥 / 2))))
22	21	rspceeqv 3586	. . . . . . . . . 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 587	. . . . . . . . 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 23157	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((^◡𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)(𝑥 / 2))) = (^◡𝐷 “ (0[,)𝑎))))
25	24	biimpar 481	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)(𝑥 / 2))) = (^◡𝐷 “ (0[,)𝑎))) → (^◡𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹)
26	5, 23, 25	syl2anc 587	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹)
27		0xr 10677	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ∈ ℝ^*)
29		rpxr 12386	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ^*)
30		0le0 11726	. . . . . . . . . . 11 ⊢ 0 ≤ 0
31	30	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 0)
32		rpre 12385	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
33	32	rehalfcld 11872	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) ∈ ℝ)
34		rphalflt 12406	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) < 𝑥)
35	33, 32, 34	ltled 10777	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) ≤ 𝑥)
36		icossico 12795	. . . . . . . . . 10 ⊢ (((0 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) ∧ (0 ≤ 0 ∧ (𝑥 / 2) ≤ 𝑥)) → (0[,)(𝑥 / 2)) ⊆ (0[,)𝑥))
37	28, 29, 31, 35, 36	syl22anc 837	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (0[,)(𝑥 / 2)) ⊆ (0[,)𝑥))
38		imass2 5932	. . . . . . . . 9 ⊢ ((0[,)(𝑥 / 2)) ⊆ (0[,)𝑥) → (^◡𝐷 “ (0[,)(𝑥 / 2))) ⊆ (^◡𝐷 “ (0[,)𝑥)))
39	17, 37, 38	3syl 18	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)(𝑥 / 2))) ⊆ (^◡𝐷 “ (0[,)𝑥)))
40		sseq1 3940	. . . . . . . . 9 ⊢ (𝑤 = (^◡𝐷 “ (0[,)(𝑥 / 2))) → (𝑤 ⊆ (^◡𝐷 “ (0[,)𝑥)) ↔ (^◡𝐷 “ (0[,)(𝑥 / 2))) ⊆ (^◡𝐷 “ (0[,)𝑥))))
41	40	rspcev 3571	. . . . . . . 8 ⊢ (((^◡𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹 ∧ (^◡𝐷 “ (0[,)(𝑥 / 2))) ⊆ (^◡𝐷 “ (0[,)𝑥))) → ∃𝑤 ∈ 𝐹 𝑤 ⊆ (^◡𝐷 “ (0[,)𝑥)))
42	26, 39, 41	syl2anc 587	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑤 ∈ 𝐹 𝑤 ⊆ (^◡𝐷 “ (0[,)𝑥)))
43		elfg 22476	. . . . . . . 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 481	. . . . . . 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 835	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹))
46		cfiluexsm 22896	. . . . . 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 1368	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑥)))
48		funimass2 6407	. . . . . . 7 ⊢ ((Fun 𝐷 ∧ (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑥))) → (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
49	48	ex 416	. . . . . 6 ⊢ (Fun 𝐷 → ((𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑥)) → (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
50	49	reximdv 3232	. . . . 5 ⊢ (Fun 𝐷 → (∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑥)) → ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
51	8, 47, 50	sylc 65	. . . 4 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
52	51	ralrimiva 3149	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹))) → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
53	4, 52	jca 515	. 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 770	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → 𝐶 ∈ (fBas‘𝑋))
55		oveq2 7143	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (0[,)𝑥) = (0[,)𝑎))
56	55	sseq2d 3947	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎)))
57	56	rexbidv 3256	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎)))
58		simp-4r 783	. . . . . . . . 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 499	. . . . . . . 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 768	. . . . . . . 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 3573	. . . . . . 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 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋))
63		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝐶 ∈ (fBas‘𝑋)
64		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦ℝ⁺
65		nfre1 3265	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
66	64, 65	nfralw 3189	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
67	63, 66	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
68	62, 67	nfan 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑦((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
69		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)
70	68, 69	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑦(((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
71		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑎 ∈ ℝ⁺
72	70, 71	nfan 1900	. . . . . . . . 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 1915	. . . . . . . . 9 ⊢ Ⅎ𝑦(^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣
74	72, 73	nfan 1900	. . . . . . . 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 731	. . . . . . . . . . . 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 22438	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦 ⊆ 𝑋)
77	75, 76	sylancom 591	. . . . . . . . . . 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 5534	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑋) → (𝑦 × 𝑦) ⊆ (𝑋 × 𝑋))
79	77, 77, 78	syl2anc 587	. . . . . . . . . 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 787	. . . . . . . . . . 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 18	. . . . . . . . . 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 3956	. . . . . . . . 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 416	. . . . . . . 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 3180	. . . . . . 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 3217	. . . . . . . 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 4142	. . . . . . . . . . . . 13 ⊢ ((𝑦 × 𝑦) ⊆ dom 𝐷 ↔ (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
87	86	biimpi 219	. . . . . . . . . . . 12 ⊢ ((𝑦 × 𝑦) ⊆ dom 𝐷 → (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
88	87	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
89		dminss 5977	. . . . . . . . . . 11 ⊢ (dom 𝐷 ∩ (𝑦 × 𝑦)) ⊆ (^◡𝐷 “ (𝐷 “ (𝑦 × 𝑦)))
90	88, 89	eqsstrrdi 3970	. . . . . . . . . 10 ⊢ (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (𝐷 “ (𝑦 × 𝑦))))
91		imass2 5932	. . . . . . . . . . 11 ⊢ ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) → (^◡𝐷 “ (𝐷 “ (𝑦 × 𝑦))) ⊆ (^◡𝐷 “ (0[,)𝑎)))
92	91	adantr 484	. . . . . . . . . 10 ⊢ (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (^◡𝐷 “ (𝐷 “ (𝑦 × 𝑦))) ⊆ (^◡𝐷 “ (0[,)𝑎)))
93	90, 92	sstrd 3925	. . . . . . . . 9 ⊢ (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)))
94	93	reximi 3206	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐶 ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)))
95	85, 94	syl 17	. . . . . . 7 ⊢ ((∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ ∀𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)))
96	61, 84, 95	syl2anc 587	. . . . . 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)))
97		r19.41v 3300	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐶 ((𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ↔ (∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
98		sstr 3923	. . . . . . . 8 ⊢ (((𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → (𝑦 × 𝑦) ⊆ 𝑣)
99	98	reximi 3206	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐶 ((𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
100	97, 99	sylbir 238	. . . . . 6 ⊢ ((∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ (^◡𝐷 “ (0[,)𝑎)) ∧ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
101	96, 100	sylancom 591	. . . . 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)
102		simp-5r 785	. . . . . . . 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))
103		simplr 768	. . . . . . . 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)
104	1	metustel 23157	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑤 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎))))
105	104	biimpa 480	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑤 ∈ 𝐹) → ∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎)))
106	102, 103, 105	syl2anc 587	. . . . . . 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)))
107		r19.41v 3300	. . . . . . . 8 ⊢ (∃𝑎 ∈ ℝ⁺ (𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣) ↔ (∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣))
108		sseq1 3940	. . . . . . . . . 10 ⊢ (𝑤 = (^◡𝐷 “ (0[,)𝑎)) → (𝑤 ⊆ 𝑣 ↔ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
109	108	biimpa 480	. . . . . . . . 9 ⊢ ((𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣) → (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
110	109	reximi 3206	. . . . . . . 8 ⊢ (∃𝑎 ∈ ℝ⁺ (𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣) → ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
111	107, 110	sylbir 238	. . . . . . 7 ⊢ ((∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎)) ∧ 𝑤 ⊆ 𝑣) → ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
112	106, 111	sylancom 591	. . . . . 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)
113	11	ad2antrr 725	. . . . . . . 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)))
114		elfg 22476	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣)))
115	114	biimpa 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣))
116	113, 115	sylancom 591	. . . . . . 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))
117	116	simprd 499	. . . . . 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)
118	112, 117	r19.29a 3248	. . . . 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)
119	101, 118	r19.29a 3248	. . . 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)
120	119	ralrimiva 3149	. . 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)
121	2	adantr 484	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
122		iscfilu 22894	. . . 4 ⊢ (((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) → (𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)∃𝑦 ∈ 𝐶 (𝑦 × 𝑦) ⊆ 𝑣)))
123	121, 122	syl 17	. . 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)))
124	54, 120, 123	mpbir2and 712	. 2 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → 𝐶 ∈ (CauFil_u‘((𝑋 × 𝑋)filGen𝐹)))
125	53, 124	impbida 800	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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ^◡ccnv 5518 dom cdm 5519 ran crn 5520 “ cima 5522 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℝ^*cxr 10663 ≤ cle 10665 / cdiv 11286 2c2 11680 ℝ⁺crp 12377 [,)cico 12728 PsMetcpsmet 20075 fBascfbas 20079 filGencfg 20080 UnifOncust 22805 CauFil_uccfilu 22892

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11688 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-psmet 20083 df-fbas 20088 df-fg 20089 df-fil 22451 df-ust 22806 df-cfilu 22893

This theorem is referenced by: cfilucfil2 23168

W3C validator