Theorem ustexsym 24448

Description: In an 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 24442	. . . . 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 24440	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ◡𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
6	1, 3, 4, 5	syl3anc 1398	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
7		ustrel 24444	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → Rel 𝑥)
8		dfrel2 6186	. . . . . . 7 ⊢ (Rel 𝑥 ↔ ◡◡𝑥 = 𝑥)
9	7, 8	sylib 221	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡◡𝑥 = 𝑥)
10	9	ineq1d 4168	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → (◡◡𝑥 ∩ ◡𝑥) = (𝑥 ∩ ◡𝑥))
11		cnvin 6139	. . . . 5 ⊢ ◡(◡𝑥 ∩ 𝑥) = (◡◡𝑥 ∩ ◡𝑥)
12		incom 4158	. . . . 5 ⊢ (◡𝑥 ∩ 𝑥) = (𝑥 ∩ ◡𝑥)
13	10, 11, 12	3eqtr4g 2822	. . . 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 4186	. . . 4 ⊢ (◡𝑥 ∩ 𝑥) ⊆ 𝑥
16		ustssco 24447	. . . . . 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 3944	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ⊆ 𝑉)
20	15, 19	sstrid 3945	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ⊆ 𝑉)
21		cnveq 5857	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → ◡𝑤 = ◡(◡𝑥 ∩ 𝑥))
22		id 23	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → 𝑤 = (◡𝑥 ∩ 𝑥))
23	21, 22	eqeq12d 2778	. . . . 5 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → (◡𝑤 = 𝑤 ↔ ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥)))
24		sseq1 3959	. . . . 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 3579	. . 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 24443	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 (𝑥 ∘ 𝑥) ⊆ 𝑉)
29	27, 28	r19.29a 3172	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 2145  ∃wrex 3088   ∩ cin 3901   ⊆ wss 3902  ^◡ccnv 5658   ∘ ccom 5663  Rel wrel 5664  ‘cfv 6537  UnifOncust 24432

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-ust 24433

This theorem is used by:  ustex2sym  24449  neipcfilu  24527