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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 5875 | . 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
2 | inss2 4206 | . 2 ⊢ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴 | |
3 | 1, 2 | eqsstri 4001 | 1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3935 ⊆ wss 3936 dom cdm 5555 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 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 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-dm 5565 df-res 5567 |
This theorem is referenced by: limsupresuz2 42010 liminfresuz2 42088 |
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