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Mirrors > Home > MPE Home > Th. List > resabs1 | Structured version Visualization version GIF version |
Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.) |
Ref | Expression |
---|---|
resabs1 | ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resres 5993 | . 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
2 | sseqin2 4214 | . . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵) | |
3 | reseq2 5975 | . . 3 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) | |
4 | 2, 3 | sylbi 216 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) |
5 | 1, 4 | eqtrid 2782 | 1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∩ cin 3946 ⊆ wss 3947 ↾ cres 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-opab 5210 df-xp 5681 df-rel 5682 df-res 5687 |
This theorem is referenced by: resabs1d 6011 resabs2 6012 resiima 6074 fun2ssres 6592 fssres2 6758 smores3 8355 setsres 17115 gsum2dlem2 19880 lindsss 21598 resthauslem 23087 ptcmpfi 23537 tsmsres 23868 ressxms 24254 nrginvrcn 24429 xrge0gsumle 24569 lebnumii 24712 dvmptresicc 25665 dfrelog 26310 relogf1o 26311 dvlog 26395 dvlog2 26397 efopnlem2 26401 wilthlem2 26809 nosupres 27446 nosupbnd2lem1 27454 noinfres 27461 noinfbnd2lem1 27469 nosupinfsep 27471 gsumle 32512 rrhre 33299 iwrdsplit 33684 rpsqrtcn 33903 pthhashvtx 34416 cvmsss2 34563 mbfposadd 36838 mzpcompact2lem 41791 eldioph2 41802 diophin 41812 diophrex 41815 2rexfrabdioph 41836 3rexfrabdioph 41837 4rexfrabdioph 41838 6rexfrabdioph 41839 7rexfrabdioph 41840 resabs1i 44135 fourierdlem46 45166 fourierdlem57 45177 fourierdlem111 45231 fouriersw 45245 psmeasurelem 45484 |
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