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Mirrors > Home > MPE Home > Th. List > ustelimasn | Structured version Visualization version GIF version |
Description: Any point 𝐴 is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.) |
Ref | Expression |
---|---|
ustelimasn | ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝑉 “ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1169 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
2 | ustdiag 22336 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉) | |
3 | 2 | 3adant3 1163 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → ( I ↾ 𝑋) ⊆ 𝑉) |
4 | opelidres 5617 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ 𝑋)) | |
5 | 4 | ibir 260 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → 〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋)) |
6 | 5 | 3ad2ant3 1166 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋)) |
7 | 3, 6 | sseldd 3797 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 〈𝐴, 𝐴〉 ∈ 𝑉) |
8 | elimasng 5706 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ 〈𝐴, 𝐴〉 ∈ 𝑉)) | |
9 | 8 | anidms 563 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ 〈𝐴, 𝐴〉 ∈ 𝑉)) |
10 | 9 | biimpar 470 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 〈𝐴, 𝐴〉 ∈ 𝑉) → 𝐴 ∈ (𝑉 “ {𝐴})) |
11 | 1, 7, 10 | syl2anc 580 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝑉 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1108 ∈ wcel 2157 ⊆ wss 3767 {csn 4366 〈cop 4372 I cid 5217 ↾ cres 5312 “ cima 5313 ‘cfv 6099 UnifOncust 22327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fv 6107 df-ust 22328 |
This theorem is referenced by: (None) |
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