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Mirrors > Home > MPE Home > Th. List > residm | Structured version Visualization version GIF version |
Description: Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.) |
Ref | Expression |
---|---|
residm | ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3966 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
2 | resabs2 5969 | . 2 ⊢ (𝐵 ⊆ 𝐵 → ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊆ wss 3910 ↾ cres 5635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-opab 5168 df-xp 5639 df-rel 5640 df-res 5645 |
This theorem is referenced by: resima 5971 dffv2 6936 fvsnun2 7128 qtopres 23047 bnj1253 33620 eldioph2lem1 41061 eldioph2lem2 41062 relexpiidm 41958 |
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