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| Mirrors > Home > MPE Home > Th. List > eldmeldmressn | Structured version Visualization version GIF version | ||
| Description: An element of the domain (of a relation) is an element of the domain of the restriction (of the relation) to the singleton containing this element. (Contributed by Alexander van der Vekens, 22-Jul-2018.) |
| Ref | Expression |
|---|---|
| eldmeldmressn | ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldmressnsn 6042 | . 2 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ↾ {𝑋})) | |
| 2 | elinel2 4202 | . . 3 ⊢ (𝑋 ∈ ({𝑋} ∩ dom 𝐹) → 𝑋 ∈ dom 𝐹) | |
| 3 | dmres 6030 | . . 3 ⊢ dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹) | |
| 4 | 2, 3 | eleq2s 2859 | . 2 ⊢ (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → 𝑋 ∈ dom 𝐹) |
| 5 | 1, 4 | impbii 209 | 1 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 ∩ cin 3950 {csn 4626 dom cdm 5685 ↾ cres 5687 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-res 5697 |
| This theorem is referenced by: (None) |
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