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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 5347 | . 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 1475 ⊆ wss 3540 ↾ cres 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 df-res 5050 |
This theorem is referenced by: f2ndf 7170 ablfac1eulem 18294 kgencn2 21170 tsmsres 21757 resubmet 22413 xrge0gsumle 22444 cmsss 22955 minveclem3a 23006 dvlip2 23562 c1liplem1 23563 efcvx 24007 logccv 24209 loglesqrt 24299 wilthlem2 24595 bnj1280 30342 cvmlift2lem9 30547 mbfresfi 32626 ssbnd 32757 prdsbnd2 32764 cnpwstotbnd 32766 reheibor 32808 diophin 36354 fnwe2lem2 36639 dvsid 37552 limcresiooub 38709 limcresioolb 38710 dvmptresicc 38809 fourierdlem46 39045 fourierdlem48 39047 fourierdlem49 39048 fourierdlem58 39057 fourierdlem72 39071 fourierdlem73 39072 fourierdlem74 39073 fourierdlem75 39074 fourierdlem89 39088 fourierdlem90 39089 fourierdlem91 39090 fourierdlem93 39092 fourierdlem100 39099 fourierdlem102 39101 fourierdlem103 39102 fourierdlem104 39103 fourierdlem107 39106 fourierdlem111 39110 fourierdlem112 39111 fourierdlem114 39113 afvres 39901 funcrngcsetc 41790 funcrngcsetcALT 41791 funcringcsetc 41827 |
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