Theorem ustssco 24039

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 4172	. . . 4 ⊢ 𝑉 ⊆ (𝑉 ∪ (𝑉 ∘ 𝑉))
2		coires1 6263	. . . . . 6 ⊢ (𝑉 ∘ ( I ↾ 𝑋)) = (𝑉 ↾ 𝑋)
3		ustrel 24036	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)
4		ustssxp 24029	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
5		dmss 5902	. . . . . . . . 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 5929	. . . . . . . 8 ⊢ dom (𝑋 × 𝑋) = 𝑋
8	6, 7	sseqtrdi 4032	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → dom 𝑉 ⊆ 𝑋)
9		relssres 6022	. . . . . . 7 ⊢ ((Rel 𝑉 ∧ dom 𝑉 ⊆ 𝑋) → (𝑉 ↾ 𝑋) = 𝑉)
10	3, 8, 9	syl2anc 583	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ↾ 𝑋) = 𝑉)
11	2, 10	eqtrid 2783	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ ( I ↾ 𝑋)) = 𝑉)
12	11	uneq1d 4162	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)) = (𝑉 ∪ (𝑉 ∘ 𝑉)))
13	1, 12	sseqtrrid 4035	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)))
14		coundi 6246	. . 3 ⊢ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉))
15	13, 14	sseqtrrdi 4033	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)))
16		ustdiag 24033	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
17		ssequn1 4180	. . . 4 ⊢ (( I ↾ 𝑋) ⊆ 𝑉 ↔ (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
18	16, 17	sylib 217	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
19	18	coeq2d 5862	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = (𝑉 ∘ 𝑉))
20	15, 19	sseqtrd 4022	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ 𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ∪ cun 3946   ⊆ wss 3948   I cid 5573   × cxp 5674  dom cdm 5676   ↾ cres 5678   ∘ ccom 5680  Rel wrel 5681  ‘cfv 6543  UnifOncust 24024

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-ust 24025

This theorem is referenced by:  ustexsym  24040  ustex3sym  24042