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Mirrors > Home > MPE Home > Th. List > ustssco | Structured version Visualization version GIF version |
Description: In an uniform structure, any entourage 𝑉 is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.) |
Ref | Expression |
---|---|
ustssco | ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4145 | . . . 4 ⊢ 𝑉 ⊆ (𝑉 ∪ (𝑉 ∘ 𝑉)) | |
2 | coires1 6110 | . . . . . 6 ⊢ (𝑉 ∘ ( I ↾ 𝑋)) = (𝑉 ↾ 𝑋) | |
3 | ustrel 22747 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉) | |
4 | ustssxp 22740 | . . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋)) | |
5 | dmss 5764 | . . . . . . . . 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 5793 | . . . . . . . 8 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
8 | 6, 7 | sseqtrdi 4014 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → dom 𝑉 ⊆ 𝑋) |
9 | relssres 5886 | . . . . . . 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 | syl5eq 2865 | . . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ ( I ↾ 𝑋)) = 𝑉) |
12 | 11 | uneq1d 4135 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)) = (𝑉 ∪ (𝑉 ∘ 𝑉))) |
13 | 1, 12 | sseqtrrid 4017 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉))) |
14 | coundi 6093 | . . 3 ⊢ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)) | |
15 | 13, 14 | sseqtrrdi 4015 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉))) |
16 | ustdiag 22744 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉) | |
17 | ssequn1 4153 | . . . 4 ⊢ (( I ↾ 𝑋) ⊆ 𝑉 ↔ (( I ↾ 𝑋) ∪ 𝑉) = 𝑉) | |
18 | 16, 17 | sylib 219 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (( I ↾ 𝑋) ∪ 𝑉) = 𝑉) |
19 | 18 | coeq2d 5726 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = (𝑉 ∘ 𝑉)) |
20 | 15, 19 | sseqtrd 4004 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 ⊆ wss 3933 I cid 5452 × cxp 5546 dom cdm 5548 ↾ cres 5550 ∘ ccom 5552 Rel wrel 5553 ‘cfv 6348 UnifOncust 22735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-ust 22736 |
This theorem is referenced by: ustexsym 22751 ustex3sym 22753 |
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