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| 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 4177 | . . . 4 ⊢ 𝑉 ⊆ (𝑉 ∪ (𝑉 ∘ 𝑉)) | |
| 2 | coires1 6283 | . . . . . 6 ⊢ (𝑉 ∘ ( I ↾ 𝑋)) = (𝑉 ↾ 𝑋) | |
| 3 | ustrel 24221 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉) | |
| 4 | ustssxp 24214 | . . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋)) | |
| 5 | dmss 5912 | . . . . . . . . 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 5940 | . . . . . . . 8 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
| 8 | 6, 7 | sseqtrdi 4023 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → dom 𝑉 ⊆ 𝑋) | 
| 9 | relssres 6039 | . . . . . . 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 2788 | . . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ ( I ↾ 𝑋)) = 𝑉) | 
| 12 | 11 | uneq1d 4166 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)) = (𝑉 ∪ (𝑉 ∘ 𝑉))) | 
| 13 | 1, 12 | sseqtrrid 4026 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉))) | 
| 14 | coundi 6266 | . . 3 ⊢ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉 ∘ 𝑉)) | |
| 15 | 13, 14 | sseqtrrdi 4024 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉))) | 
| 16 | ustdiag 24218 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉) | |
| 17 | ssequn1 4185 | . . . 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 5872 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = (𝑉 ∘ 𝑉)) | 
| 20 | 15, 19 | sseqtrd 4019 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ 𝑉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∪ cun 3948 ⊆ wss 3950 I cid 5576 × cxp 5682 dom cdm 5684 ↾ cres 5686 ∘ ccom 5688 Rel wrel 5689 ‘cfv 6560 UnifOncust 24209 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fv 6568 df-ust 24210 | 
| This theorem is referenced by: ustexsym 24225 ustex3sym 24227 | 
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