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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 5866 | . 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
2 | sseqin2 4192 | . . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵) | |
3 | reseq2 5848 | . . 3 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) | |
4 | 2, 3 | sylbi 219 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) |
5 | 1, 4 | syl5eq 2868 | 1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3935 ⊆ wss 3936 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-xp 5561 df-rel 5562 df-res 5567 |
This theorem is referenced by: resabs1d 5884 resabs2 5885 resiima 5944 fun2ssres 6399 fssres2 6546 smores3 7990 setsres 16525 gsum2dlem2 19091 lindsss 20968 resthauslem 21971 ptcmpfi 22421 tsmsres 22752 ressxms 23135 nrginvrcn 23301 xrge0gsumle 23441 lebnumii 23570 dfrelog 25149 relogf1o 25150 dvlog 25234 dvlog2 25236 efopnlem2 25240 wilthlem2 25646 gsumle 30725 rrhre 31262 iwrdsplit 31645 rpsqrtcn 31864 pthhashvtx 32374 cvmsss2 32521 nosupres 33207 nosupbnd2lem1 33215 mbfposadd 34954 mzpcompact2lem 39368 eldioph2 39379 diophin 39389 diophrex 39392 2rexfrabdioph 39413 3rexfrabdioph 39414 4rexfrabdioph 39415 6rexfrabdioph 39416 7rexfrabdioph 39417 resabs1i 41434 dvmptresicc 42224 fourierdlem46 42457 fourierdlem57 42468 fourierdlem111 42522 fouriersw 42536 psmeasurelem 42772 |
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