Theorem ustexsym 24568

Description: In a uniform structure, for any entourage 𝑉, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)

Assertion

Ref	Expression
ustexsym	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))

Distinct variable groups:   𝑤,𝑈   𝑤,𝑉

Allowed substitution hint:   𝑋(𝑤)

Proof of Theorem ustexsym

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplll 787	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2		ustinvel 24562	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡𝑥 ∈ 𝑈)
3	2	ad4ant13 764	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ◡𝑥 ∈ 𝑈)
4		simplr 781	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ∈ 𝑈)
5		ustincl 24560	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ◡𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
6	1, 3, 4, 5	syl3anc 1399	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
7		ustrel 24564	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → Rel 𝑥)
8		dfrel2 6194	. . . . . . 7 ⊢ (Rel 𝑥 ↔ ◡◡𝑥 = 𝑥)
9	7, 8	sylib 221	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡◡𝑥 = 𝑥)
10	9	ineq1d 4119	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → (◡◡𝑥 ∩ ◡𝑥) = (𝑥 ∩ ◡𝑥))
11		cnvin 6148	. . . . 5 ⊢ ◡(◡𝑥 ∩ 𝑥) = (◡◡𝑥 ∩ ◡𝑥)
12		incom 4109	. . . . 5 ⊢ (◡𝑥 ∩ 𝑥) = (𝑥 ∩ ◡𝑥)
13	10, 11, 12	3eqtr4g 2823	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥))
14	13	ad4ant13 764	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥))
15		inss2 4137	. . . 4 ⊢ (◡𝑥 ∩ 𝑥) ⊆ 𝑥
16		ustssco 24567	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ (𝑥 ∘ 𝑥))
17	16	ad4ant13 764	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ⊆ (𝑥 ∘ 𝑥))
18		simpr 490	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (𝑥 ∘ 𝑥) ⊆ 𝑉)
19	17, 18	sstrd 3923	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ⊆ 𝑉)
20	15, 19	sstrid 3924	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ⊆ 𝑉)
21		cnveq 5859	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → ◡𝑤 = ◡(◡𝑥 ∩ 𝑥))
22		id 23	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → 𝑤 = (◡𝑥 ∩ 𝑥))
23	21, 22	eqeq12d 2779	. . . . 5 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → (◡𝑤 = 𝑤 ↔ ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥)))
24		sseq1 3938	. . . . 5 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → (𝑤 ⊆ 𝑉 ↔ (◡𝑥 ∩ 𝑥) ⊆ 𝑉))
25	23, 24	anbi12d 644	. . . 4 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → ((◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉) ↔ (◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥) ∧ (◡𝑥 ∩ 𝑥) ⊆ 𝑉)))
26	25	rspcev 3580	. . 3 ⊢ (((◡𝑥 ∩ 𝑥) ∈ 𝑈 ∧ (◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥) ∧ (◡𝑥 ∩ 𝑥) ⊆ 𝑉)) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
27	6, 14, 20, 26	syl12anc 850	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
28		ustexhalf 24563	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 (𝑥 ∘ 𝑥) ⊆ 𝑉)
29	27, 28	r19.29a 3174	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  ^◡ccnv 5655   ∘ ccom 5660  Rel wrel 5661  ‘cfv 6561  UnifOncust 24552

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6515  df-fun 6563  df-fv 6569  df-ust 24553

This theorem is used by:  ustex2sym  24569  neipcfilu  24647