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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 4565 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ {𝐴}) | |
2 | dmressnsn 5882 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴}) | |
3 | 1, 2 | eleqtrrd 2837 | 1 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 {csn 4531 dom cdm 5540 ↾ cres 5542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-dm 5550 df-res 5552 |
This theorem is referenced by: eldmeldmressn 5884 fvn0fvelrn 6967 funressndmfvrn 44164 dfdfat2 44246 |
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