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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmresss | Structured version Visualization version GIF version |
Description: The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
dmresss | ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 5913 | . 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
2 | inss2 4163 | . 2 ⊢ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴 | |
3 | 1, 2 | eqsstri 3955 | 1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3886 ⊆ wss 3887 dom cdm 5589 ↾ cres 5591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-dm 5599 df-res 5601 |
This theorem is referenced by: limsupresuz2 43250 liminfresuz2 43328 |
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