Theorem ustexsym 24181

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 775	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2		ustinvel 24175	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡𝑥 ∈ 𝑈)
3	2	ad4ant13 752	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ◡𝑥 ∈ 𝑈)
4		simplr 769	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ∈ 𝑈)
5		ustincl 24173	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ◡𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
6	1, 3, 4, 5	syl3anc 1374	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ∈ 𝑈)
7		ustrel 24177	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → Rel 𝑥)
8		dfrel2 6153	. . . . . . 7 ⊢ (Rel 𝑥 ↔ ◡◡𝑥 = 𝑥)
9	7, 8	sylib 218	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡◡𝑥 = 𝑥)
10	9	ineq1d 4159	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → (◡◡𝑥 ∩ ◡𝑥) = (𝑥 ∩ ◡𝑥))
11		cnvin 6108	. . . . 5 ⊢ ◡(◡𝑥 ∩ 𝑥) = (◡◡𝑥 ∩ ◡𝑥)
12		incom 4149	. . . . 5 ⊢ (◡𝑥 ∩ 𝑥) = (𝑥 ∩ ◡𝑥)
13	10, 11, 12	3eqtr4g 2796	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥))
14	13	ad4ant13 752	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥))
15		inss2 4178	. . . 4 ⊢ (◡𝑥 ∩ 𝑥) ⊆ 𝑥
16		ustssco 24180	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ (𝑥 ∘ 𝑥))
17	16	ad4ant13 752	. . . . 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 3932	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → 𝑥 ⊆ 𝑉)
20	15, 19	sstrid 3933	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → (◡𝑥 ∩ 𝑥) ⊆ 𝑉)
21		cnveq 5828	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → ◡𝑤 = ◡(◡𝑥 ∩ 𝑥))
22		id 22	. . . . . 6 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → 𝑤 = (◡𝑥 ∩ 𝑥))
23	21, 22	eqeq12d 2752	. . . . 5 ⊢ (𝑤 = (◡𝑥 ∩ 𝑥) → (◡𝑤 = 𝑤 ↔ ◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥)))
24		sseq1 3947	. . . . 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 3564	. . 3 ⊢ (((◡𝑥 ∩ 𝑥) ∈ 𝑈 ∧ (◡(◡𝑥 ∩ 𝑥) = (◡𝑥 ∩ 𝑥) ∧ (◡𝑥 ∩ 𝑥) ⊆ 𝑉)) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
27	6, 14, 20, 26	syl12anc 837	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
28		ustexhalf 24176	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 (𝑥 ∘ 𝑥) ⊆ 𝑉)
29	27, 28	r19.29a 3145	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  ^◡ccnv 5630   ∘ ccom 5635  Rel wrel 5636  ‘cfv 6498  UnifOncust 24165

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6454  df-fun 6500  df-fv 6506  df-ust 24166

This theorem is referenced by:  ustex2sym  24182  neipcfilu  24260