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Mirrors > Home > MPE Home > Th. List > eldmressnsn | Structured version Visualization version GIF version |
Description: The element of the domain of a restriction to a singleton is the element of the singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
Ref | Expression |
---|---|
eldmressnsn | ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snidg 4665 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ {𝐴}) | |
2 | dmressnsn 6043 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴}) | |
3 | 1, 2 | eleqtrrd 2842 | 1 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 {csn 4631 dom cdm 5689 ↾ cres 5691 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 df-res 5701 |
This theorem is referenced by: eldmeldmressn 6045 fvn0fvelrnOLD 7183 funressndmfvrn 46994 dfdfat2 47078 |
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