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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 5955 | . 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
2 | sseqin2 4180 | . . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵) | |
3 | reseq2 5937 | . . 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 2789 | 1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∩ cin 3914 ⊆ wss 3915 ↾ cres 5640 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-opab 5173 df-xp 5644 df-rel 5645 df-res 5650 |
This theorem is referenced by: resabs1d 5973 resabs2 5974 resiima 6033 fun2ssres 6551 fssres2 6715 smores3 8304 setsres 17057 gsum2dlem2 19755 lindsss 21246 resthauslem 22730 ptcmpfi 23180 tsmsres 23511 ressxms 23897 nrginvrcn 24072 xrge0gsumle 24212 lebnumii 24345 dvmptresicc 25296 dfrelog 25937 relogf1o 25938 dvlog 26022 dvlog2 26024 efopnlem2 26028 wilthlem2 26434 nosupres 27071 nosupbnd2lem1 27079 noinfres 27086 noinfbnd2lem1 27094 nosupinfsep 27096 gsumle 31974 rrhre 32642 iwrdsplit 33027 rpsqrtcn 33246 pthhashvtx 33761 cvmsss2 33908 mbfposadd 36154 mzpcompact2lem 41103 eldioph2 41114 diophin 41124 diophrex 41127 2rexfrabdioph 41148 3rexfrabdioph 41149 4rexfrabdioph 41150 6rexfrabdioph 41151 7rexfrabdioph 41152 resabs1i 43429 fourierdlem46 44467 fourierdlem57 44478 fourierdlem111 44532 fouriersw 44546 psmeasurelem 44785 |
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