Theorem ustssco 24351

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 4130	. . . 4 ⊢ 𝑉 ⊆ (𝑉 ∪ (𝑉 ∘ 𝑉))
2		coires1 6266	. . . . . 6 ⊢ (𝑉 ∘ ( I ↾ 𝑋)) = (𝑉 ↾ 𝑋)
3		ustrel 24348	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)
4		ustssxp 24341	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
5		dmss 5892	. . . . . . . . 9 ⊢ (𝑉 ⊆ (𝑋 × 𝑋) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
6	4, 5	syl 18	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
7		dmxpid 5920	. . . . . . . 8 ⊢ dom (𝑋 × 𝑋) = 𝑋
8	6, 7	sseqtrdi 3976	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → dom 𝑉 ⊆ 𝑋)
9		relssres 6021	. . . . . . 7 ⊢ ((Rel 𝑉 ∧ dom 𝑉 ⊆ 𝑋) → (𝑉 ↾ 𝑋) = 𝑉)
10	3, 8, 9	syl2anc 595	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ↾ 𝑋) = 𝑉)
11	2, 10	eqtrid 2808	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ ( I ↾ 𝑋)) = 𝑉)
12	11	uneq1d 4120	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)) = (𝑉 ∪ (𝑉 ∘ 𝑉)))
13	1, 12	sseqtrrid 3979	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)))
14		coundi 6248	. . 3 ⊢ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉))
15	13, 14	sseqtrrdi 3977	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)))
16		ustdiag 24345	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
17		ssequn1 4138	. . . 4 ⊢ (( I ↾ 𝑋) ⊆ 𝑉 ↔ (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
18	16, 17	sylib 221	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
19	18	coeq2d 5848	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = (𝑉 ∘ 𝑉))
20	15, 19	sseqtrd 3972	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ 𝑉))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ∪ cun 3902   ⊆ wss 3904   I cid 5555   × cxp 5659  dom cdm 5661   ↾ cres 5663   ∘ ccom 5665  Rel wrel 5666  ‘cfv 6536  UnifOncust 24336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ust 24337

This theorem is referenced by:  ustexsym  24352  ustex3sym  24354