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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 1130 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
2 | ustdiag 22744 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉) | |
3 | 2 | 3adant3 1124 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → ( I ↾ 𝑋) ⊆ 𝑉) |
4 | opelidres 5858 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ 𝑋)) | |
5 | 4 | ibir 269 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → 〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋)) |
6 | 5 | 3ad2ant3 1127 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋)) |
7 | 3, 6 | sseldd 3965 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 〈𝐴, 𝐴〉 ∈ 𝑉) |
8 | elimasng 5948 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ 〈𝐴, 𝐴〉 ∈ 𝑉)) | |
9 | 8 | anidms 567 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ 〈𝐴, 𝐴〉 ∈ 𝑉)) |
10 | 9 | biimpar 478 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 〈𝐴, 𝐴〉 ∈ 𝑉) → 𝐴 ∈ (𝑉 “ {𝐴})) |
11 | 1, 7, 10 | syl2anc 584 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝑉 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 ∈ wcel 2105 ⊆ wss 3933 {csn 4557 〈cop 4563 I cid 5452 ↾ cres 5550 “ cima 5551 ‘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-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-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ust 22736 |
This theorem is referenced by: (None) |
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