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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 1134 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
2 | ustdiag 22811 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉) | |
3 | 2 | 3adant3 1128 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → ( I ↾ 𝑋) ⊆ 𝑉) |
4 | opelidres 5859 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ 𝑋)) | |
5 | 4 | ibir 270 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → 〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋)) |
6 | 5 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋)) |
7 | 3, 6 | sseldd 3967 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 〈𝐴, 𝐴〉 ∈ 𝑉) |
8 | elimasng 5949 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ 〈𝐴, 𝐴〉 ∈ 𝑉)) | |
9 | 8 | anidms 569 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ 〈𝐴, 𝐴〉 ∈ 𝑉)) |
10 | 9 | biimpar 480 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 〈𝐴, 𝐴〉 ∈ 𝑉) → 𝐴 ∈ (𝑉 “ {𝐴})) |
11 | 1, 7, 10 | syl2anc 586 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝑉 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2110 ⊆ wss 3935 {csn 4560 〈cop 4566 I cid 5453 ↾ cres 5551 “ cima 5552 ‘cfv 6349 UnifOncust 22802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fv 6357 df-ust 22803 |
This theorem is referenced by: (None) |
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