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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 1056 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
2 | ustdiag 21822 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉) | |
3 | 2 | 3adant3 1074 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → ( I ↾ 𝑋) ⊆ 𝑉) |
4 | opelresi 5328 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ 𝑋)) | |
5 | 4 | ibir 256 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → 〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋)) |
6 | 5 | 3ad2ant3 1077 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 〈𝐴, 𝐴〉 ∈ ( I ↾ 𝑋)) |
7 | 3, 6 | sseldd 3569 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 〈𝐴, 𝐴〉 ∈ 𝑉) |
8 | elimasng 5410 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ 〈𝐴, 𝐴〉 ∈ 𝑉)) | |
9 | 8 | anidms 675 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ 〈𝐴, 𝐴〉 ∈ 𝑉)) |
10 | 9 | biimpar 501 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 〈𝐴, 𝐴〉 ∈ 𝑉) → 𝐴 ∈ (𝑉 “ {𝐴})) |
11 | 1, 7, 10 | syl2anc 691 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝑉 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 〈cop 4131 I cid 4948 ↾ cres 5040 “ cima 5041 ‘cfv 5804 UnifOncust 21813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ust 21814 |
This theorem is referenced by: (None) |
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