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Mirrors > Home > MPE Home > Th. List > Mathboxes > resabs1i | Structured version Visualization version GIF version |
Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
resabs1i.1 | ⊢ 𝐵 ⊆ 𝐶 |
Ref | Expression |
---|---|
resabs1i | ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resabs1i.1 | . 2 ⊢ 𝐵 ⊆ 𝐶 | |
2 | resabs1 5876 | . 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ⊆ wss 3933 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-rel 5555 df-res 5560 |
This theorem is referenced by: liminfresre 41936 |
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