Theorem ustex2sym 22808

Description: In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than half 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)

Assertion

Ref	Expression
ustex2sym	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉))

Distinct variable groups:   𝑤,𝑈   𝑤,𝑉   𝑤,𝑋

Proof of Theorem ustex2sym

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ustexsym 22807	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣))
2	1	ad4ant13 749	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣))
3		simprl 769	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → ◡𝑤 = 𝑤)
4		coss1 5712	. . . . . . . . 9 ⊢ (𝑤 ⊆ 𝑣 → (𝑤 ∘ 𝑤) ⊆ (𝑣 ∘ 𝑤))
5		coss2 5713	. . . . . . . . 9 ⊢ (𝑤 ⊆ 𝑣 → (𝑣 ∘ 𝑤) ⊆ (𝑣 ∘ 𝑣))
6	4, 5	sstrd 3965	. . . . . . . 8 ⊢ (𝑤 ⊆ 𝑣 → (𝑤 ∘ 𝑤) ⊆ (𝑣 ∘ 𝑣))
7	6	ad2antll 727	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑤 ∘ 𝑤) ⊆ (𝑣 ∘ 𝑣))
8		simpllr 774	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑣 ∘ 𝑣) ⊆ 𝑉)
9	7, 8	sstrd 3965	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑤 ∘ 𝑤) ⊆ 𝑉)
10	3, 9	jca 514	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉))
11	10	ex 415	. . . 4 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) → ((◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣) → (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉)))
12	11	reximdva 3274	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → (∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉)))
13	2, 12	mpd 15	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉))
14		ustexhalf 22802	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑣 ∈ 𝑈 (𝑣 ∘ 𝑣) ⊆ 𝑉)
15	13, 14	r19.29a 3289	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3139   ⊆ wss 3924  ^◡ccnv 5540   ∘ ccom 5545  ‘cfv 6341  UnifOncust 22791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3488  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-iota 6300  df-fun 6343  df-fv 6349  df-ust 22792

This theorem is referenced by:  ustex3sym  22809