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Mirrors > Home > MPE Home > Th. List > dmressnsn | Structured version Visualization version GIF version |
Description: The domain of a restriction to a singleton is a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
Ref | Expression |
---|---|
dmressnsn | ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 5869 | . 2 ⊢ dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹) | |
2 | snssi 4734 | . . 3 ⊢ (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹) | |
3 | df-ss 3951 | . . 3 ⊢ ({𝐴} ⊆ dom 𝐹 ↔ ({𝐴} ∩ dom 𝐹) = {𝐴}) | |
4 | 2, 3 | sylib 220 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → ({𝐴} ∩ dom 𝐹) = {𝐴}) |
5 | 1, 4 | syl5eq 2868 | 1 ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 {csn 4560 dom cdm 5549 ↾ cres 5551 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-dm 5559 df-res 5561 |
This theorem is referenced by: eldmressnsn 5889 funcoressn 43271 funressnfv 43272 |
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