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Mirrors > Home > MPE Home > Th. List > resabs1d | Structured version Visualization version GIF version |
Description: Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
resabs1d.b | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
resabs1d | ⊢ (𝜑 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resabs1d.b | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
2 | resabs1 5427 | . 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ⊆ 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: f2ndf 7283 ablfac1eulem 18471 kgencn2 21360 tsmsres 21947 resubmet 22605 xrge0gsumle 22636 cmsss 23147 minveclem3a 23198 dvlip2 23758 c1liplem1 23759 efcvx 24203 logccv 24409 loglesqrt 24499 wilthlem2 24795 bnj1280 31088 cvmlift2lem9 31293 nosupno 31849 nosupbnd1lem1 31854 nosupbnd2 31862 mbfresfi 33456 ssbnd 33587 prdsbnd2 33594 cnpwstotbnd 33596 reheibor 33638 diophin 37336 fnwe2lem2 37621 dvsid 38530 limcresiooub 39874 limcresioolb 39875 dvmptresicc 40134 fourierdlem46 40369 fourierdlem48 40371 fourierdlem49 40372 fourierdlem58 40381 fourierdlem72 40395 fourierdlem73 40396 fourierdlem74 40397 fourierdlem75 40398 fourierdlem89 40412 fourierdlem90 40413 fourierdlem91 40414 fourierdlem93 40416 fourierdlem100 40423 fourierdlem102 40425 fourierdlem103 40426 fourierdlem104 40427 fourierdlem107 40430 fourierdlem111 40434 fourierdlem112 40435 fourierdlem114 40437 afvres 41252 funcrngcsetc 41998 funcrngcsetcALT 41999 funcringcsetc 42035 |
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