Theorem ustexsym 24103

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 774	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2		ustinvel 24097	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡𝑥 ∈ 𝑈)
3	2	ad4ant13 751	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ◡𝑥 ∈ 𝑈)
4		simplr 768	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ∈ 𝑈)
5		ustincl 24095	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ◡𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
6	1, 3, 4, 5	syl3anc 1373	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
7		ustrel 24099	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → Rel 𝑥)
8		dfrel2 6162	. . . . . . 7 ⊢ (Rel 𝑥 ↔ ◡◡𝑥 = 𝑥)
9	7, 8	sylib 218	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡◡𝑥 = 𝑥)
10	9	ineq1d 4182	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → (◡◡𝑥 ∩ ◡𝑥) = (𝑥 ∩ ◡𝑥))
11		cnvin 6117	. . . . 5 ⊢ ◡(◡𝑥 ∩ 𝑥) = (◡◡𝑥 ∩ ◡𝑥)
12		incom 4172	. . . . 5 ⊢ (◡𝑥 ∩ 𝑥) = (𝑥 ∩ ◡𝑥)
13	10, 11, 12	3eqtr4g 2789	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥))
14	13	ad4ant13 751	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥))
15		inss2 4201	. . . 4 ⊢ (◡𝑥 ∩ 𝑥) ⊆ 𝑥
16		ustssco 24102	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ (𝑥 ∘ 𝑥))
17	16	ad4ant13 751	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ⊆ (𝑥 ∘ 𝑥))
18		simpr 484	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (𝑥 ∘ 𝑥) ⊆ 𝑉)
19	17, 18	sstrd 3957	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ⊆ 𝑉)
20	15, 19	sstrid 3958	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ⊆ 𝑉)
21		cnveq 5837	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → ◡𝑤 = ◡(◡𝑥 ∩ 𝑥))
22		id 22	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → 𝑤 = (◡𝑥 ∩ 𝑥))
23	21, 22	eqeq12d 2745	. . . . 5 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → (◡𝑤 = 𝑤 ↔ ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥)))
24		sseq1 3972	. . . . 5 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → (𝑤 ⊆ 𝑉 ↔ (◡𝑥 ∩ 𝑥) ⊆ 𝑉))
25	23, 24	anbi12d 632	. . . 4 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → ((◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉) ↔ (◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥) ∧ (◡𝑥 ∩ 𝑥) ⊆ 𝑉)))
26	25	rspcev 3588	. . 3 ⊢ (((◡𝑥 ∩ 𝑥) ∈ 𝑈 ∧ (◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥) ∧ (◡𝑥 ∩ 𝑥) ⊆ 𝑉)) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
27	6, 14, 20, 26	syl12anc 836	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
28		ustexhalf 24098	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 (𝑥 ∘ 𝑥) ⊆ 𝑉)
29	27, 28	r19.29a 3141	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∩ cin 3913   ⊆ wss 3914  ^◡ccnv 5637   ∘ ccom 5642  Rel wrel 5643  ‘cfv 6511  UnifOncust 24087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fv 6519  df-ust 24088

This theorem is referenced by:  ustex2sym  24104  neipcfilu  24183