Theorem cfiluexsm 24214

Description: For a Cauchy filter base and any entourage 𝑉, there is an element of the filter small in 𝑉. (Contributed by Thierry Arnoux, 19-Nov-2017.)

Assertion

Ref	Expression
cfiluexsm	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈) ∧ 𝑉 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑉)

Distinct variable groups:   𝐹,𝑎   𝑉,𝑎

Allowed substitution hints:   𝑈(𝑎)   𝑋(𝑎)

Proof of Theorem cfiluexsm

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscfilu 24212	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
2	1	simplbda 499	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3	2	3adant3 1132	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈) ∧ 𝑉 ∈ 𝑈) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
4		sseq2 3958	. . . . 5 ⊢ (𝑣 = 𝑉 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑎 × 𝑎) ⊆ 𝑉))
5	4	rexbidv 3158	. . . 4 ⊢ (𝑣 = 𝑉 → (∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
6	5	rspcv 3570	. . 3 ⊢ (𝑉 ∈ 𝑈 → (∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣 → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
7	6	3ad2ant3 1135	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈) ∧ 𝑉 ∈ 𝑈) → (∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣 → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
8	3, 7	mpd 15	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈) ∧ 𝑉 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑉)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058   ⊆ wss 3899   × cxp 5619  ‘cfv 6489  fBascfbas 21289  UnifOncust 24125  CauFil_uccfilu 24210

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-iota 6445  df-fun 6491  df-fv 6497  df-ust 24126  df-cfilu 24211

This theorem is referenced by:  fmucnd  24216  cfilucfil  24484