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| Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmressn | Structured version Visualization version GIF version | ||
| Description: Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.) | 
| Ref | Expression | 
|---|---|
| eldmressn | ⊢ (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elin 3966 | . . 3 ⊢ (𝐵 ∈ ({𝐴} ∩ dom 𝐹) ↔ (𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹)) | |
| 2 | elsni 4642 | . . . 4 ⊢ (𝐵 ∈ {𝐴} → 𝐵 = 𝐴) | |
| 3 | 2 | adantr 480 | . . 3 ⊢ ((𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹) → 𝐵 = 𝐴) | 
| 4 | 1, 3 | sylbi 217 | . 2 ⊢ (𝐵 ∈ ({𝐴} ∩ dom 𝐹) → 𝐵 = 𝐴) | 
| 5 | dmres 6029 | . 2 ⊢ dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹) | |
| 6 | 4, 5 | eleq2s 2858 | 1 ⊢ (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∩ cin 3949 {csn 4625 dom cdm 5684 ↾ cres 5686 | 
| 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-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 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-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-dm 5694 df-res 5696 | 
| This theorem is referenced by: dfdfat2 47145 | 
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