Theorem ustssco 24239

Description: In an uniform structure, any entourage 𝑉 is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)

Assertion

Ref	Expression
ustssco	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ 𝑉))

Proof of Theorem ustssco

Step	Hyp	Ref	Expression
1		ssun1 4188	. . . 4 ⊢ 𝑉 ⊆ (𝑉 ∪ (𝑉 ∘ 𝑉))
2		coires1 6286	. . . . . 6 ⊢ (𝑉 ∘ ( I ↾ 𝑋)) = (𝑉 ↾ 𝑋)
3		ustrel 24236	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)
4		ustssxp 24229	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
5		dmss 5916	. . . . . . . . 9 ⊢ (𝑉 ⊆ (𝑋 × 𝑋) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
7		dmxpid 5944	. . . . . . . 8 ⊢ dom (𝑋 × 𝑋) = 𝑋
8	6, 7	sseqtrdi 4046	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → dom 𝑉 ⊆ 𝑋)
9		relssres 6042	. . . . . . 7 ⊢ ((Rel 𝑉 ∧ dom 𝑉 ⊆ 𝑋) → (𝑉 ↾ 𝑋) = 𝑉)
10	3, 8, 9	syl2anc 584	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ↾ 𝑋) = 𝑉)
11	2, 10	eqtrid 2787	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ ( I ↾ 𝑋)) = 𝑉)
12	11	uneq1d 4177	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)) = (𝑉 ∪ (𝑉 ∘ 𝑉)))
13	1, 12	sseqtrrid 4049	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)))
14		coundi 6269	. . 3 ⊢ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉))
15	13, 14	sseqtrrdi 4047	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)))
16		ustdiag 24233	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
17		ssequn1 4196	. . . 4 ⊢ (( I ↾ 𝑋) ⊆ 𝑉 ↔ (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
18	16, 17	sylib 218	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
19	18	coeq2d 5876	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = (𝑉 ∘ 𝑉))
20	15, 19	sseqtrd 4036	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ 𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ∪ cun 3961   ⊆ wss 3963   I cid 5582   × cxp 5687  dom cdm 5689   ↾ cres 5691   ∘ ccom 5693  Rel wrel 5694  ‘cfv 6563  UnifOncust 24224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-ust 24225

This theorem is referenced by:  ustexsym  24240  ustex3sym  24242