Theorem ustexsym 22929

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 22923	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡𝑥 ∈ 𝑈)
3	2	ad4ant13 750	. . . 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 22921	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ◡𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
6	1, 3, 4, 5	syl3anc 1368	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
7		ustrel 22925	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → Rel 𝑥)
8		dfrel2 6023	. . . . . . 7 ⊢ (Rel 𝑥 ↔ ◡◡𝑥 = 𝑥)
9	7, 8	sylib 221	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡◡𝑥 = 𝑥)
10	9	ineq1d 4118	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → (◡◡𝑥 ∩ ◡𝑥) = (𝑥 ∩ ◡𝑥))
11		cnvin 5980	. . . . 5 ⊢ ◡(◡𝑥 ∩ 𝑥) = (◡◡𝑥 ∩ ◡𝑥)
12		incom 4108	. . . . 5 ⊢ (◡𝑥 ∩ 𝑥) = (𝑥 ∩ ◡𝑥)
13	10, 11, 12	3eqtr4g 2818	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥))
14	13	ad4ant13 750	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥))
15		inss2 4136	. . . 4 ⊢ (◡𝑥 ∩ 𝑥) ⊆ 𝑥
16		ustssco 22928	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ (𝑥 ∘ 𝑥))
17	16	ad4ant13 750	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ⊆ (𝑥 ∘ 𝑥))
18		simpr 488	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (𝑥 ∘ 𝑥) ⊆ 𝑉)
19	17, 18	sstrd 3904	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ⊆ 𝑉)
20	15, 19	sstrid 3905	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ⊆ 𝑉)
21		cnveq 5719	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → ◡𝑤 = ◡(◡𝑥 ∩ 𝑥))
22		id 22	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → 𝑤 = (◡𝑥 ∩ 𝑥))
23	21, 22	eqeq12d 2774	. . . . 5 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → (◡𝑤 = 𝑤 ↔ ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥)))
24		sseq1 3919	. . . . 5 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → (𝑤 ⊆ 𝑉 ↔ (◡𝑥 ∩ 𝑥) ⊆ 𝑉))
25	23, 24	anbi12d 633	. . . 4 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → ((◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉) ↔ (◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥) ∧ (◡𝑥 ∩ 𝑥) ⊆ 𝑉)))
26	25	rspcev 3543	. . 3 ⊢ (((◡𝑥 ∩ 𝑥) ∈ 𝑈 ∧ (◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥) ∧ (◡𝑥 ∩ 𝑥) ⊆ 𝑉)) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
27	6, 14, 20, 26	syl12anc 835	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
28		ustexhalf 22924	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 (𝑥 ∘ 𝑥) ⊆ 𝑉)
29	27, 28	r19.29a 3213	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ∩ cin 3859   ⊆ wss 3860  ^◡ccnv 5527   ∘ ccom 5532  Rel wrel 5533  ‘cfv 6340  UnifOncust 22913

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 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6299  df-fun 6342  df-fv 6348  df-ust 22914

This theorem is referenced by:  ustex2sym  22930  neipcfilu  23010