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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 5409 | . 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
2 | sseqin2 3817 | . . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵) | |
3 | reseq2 5391 | . . 3 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) | |
4 | 2, 3 | sylbi 207 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) |
5 | 1, 4 | syl5eq 2668 | 1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∩ cin 3573 ⊆ wss 3574 ↾ cres 5116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 df-rel 5121 df-res 5126 |
This theorem is referenced by: resabs1d 5428 resabs2 5429 resiima 5480 fun2ssres 5931 fssres2 6072 smores3 7450 setsres 15901 gsum2dlem2 18370 lindsss 20163 resthauslem 21167 ptcmpfi 21616 tsmsres 21947 ressxms 22330 nrginvrcn 22496 xrge0gsumle 22636 lebnumii 22765 dfrelog 24312 relogf1o 24313 dvlog 24397 dvlog2 24399 efopnlem2 24403 wilthlem2 24795 gsumle 29779 rrhre 30065 iwrdsplit 30449 rpsqrtcn 30671 cvmsss2 31256 nosupres 31853 nosupbnd2lem1 31861 mbfposadd 33457 mzpcompact2lem 37314 eldioph2 37325 diophin 37336 diophrex 37339 2rexfrabdioph 37360 3rexfrabdioph 37361 4rexfrabdioph 37362 6rexfrabdioph 37363 7rexfrabdioph 37364 resabs1i 39336 dvmptresicc 40134 fourierdlem46 40369 fourierdlem57 40380 fourierdlem111 40434 fouriersw 40448 psmeasurelem 40687 |
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