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Mirrors > Home > MPE Home > Th. List > ressabs | Structured version Visualization version GIF version |
Description: Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
ressabs | ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 4956 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑋) → 𝐵 ∈ V) | |
2 | 1 | ancoms 468 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V) |
3 | ressress 16160 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ V) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))) | |
4 | 2, 3 | syldan 488 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))) |
5 | simpr 479 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴) | |
6 | sseqin2 3960 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
7 | 5, 6 | sylib 208 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∩ 𝐵) = 𝐵) |
8 | 7 | oveq2d 6830 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s 𝐵)) |
9 | 4, 8 | eqtrd 2794 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∩ cin 3714 ⊆ wss 3715 (class class class)co 6814 ↾s cress 16080 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-i2m1 10216 ax-1ne0 10217 ax-rrecex 10220 ax-cnre 10221 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-nn 11233 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 |
This theorem is referenced by: rescabs 16714 rescabs2 16715 subsubm 17578 subsubg 17838 subgslw 18251 pgpfaclem1 18700 ablfaclem3 18706 subsubrg 19028 lsslss 19183 xrge0cmn 20010 zringunit 20058 cnmsgngrp 20147 psgninv 20150 zrhpsgnmhm 20152 xrge0gsumle 22857 xrge0tsms 22858 reefgim 24423 xrge0tsmsd 30115 nn0omnd 30171 nn0archi 30173 rrhcn 30371 qqtopn 30385 lnmlsslnm 38171 lmhmlnmsplit 38177 gsumge0cl 41109 sge0tsms 41118 subsubmgm 42325 amgmlemALT 43080 |
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