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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 5118 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑋) → 𝐵 ∈ V) | |
2 | 1 | ancoms 459 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V) |
3 | ressress 16391 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ V) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))) | |
4 | 2, 3 | syldan 591 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))) |
5 | simpr 485 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴) | |
6 | sseqin2 4112 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
7 | 5, 6 | sylib 219 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∩ 𝐵) = 𝐵) |
8 | 7 | oveq2d 7032 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s 𝐵)) |
9 | 4, 8 | eqtrd 2831 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∩ cin 3858 ⊆ wss 3859 (class class class)co 7016 ↾s cress 16313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-1cn 10441 ax-addcl 10443 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-nn 11487 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 |
This theorem is referenced by: rescabs 16932 rescabs2 16933 subsubm 17796 subsubg 18056 subgslw 18471 pgpfaclem1 18920 ablfaclem3 18926 subsubrg 19251 subdrgint 19272 lsslss 19423 xrge0cmn 20269 zringunit 20317 cnmsgngrp 20405 psgninv 20408 zrhpsgnmhm 20410 xrge0gsumle 23124 xrge0tsms 23125 reefgim 24721 xrge0tsmsd 30503 nn0omnd 30568 nn0archi 30570 fedgmullem1 30629 fedgmullem2 30630 fedgmul 30631 rrhcn 30855 qqtopn 30869 lnmlsslnm 39166 lmhmlnmsplit 39172 gsumge0cl 42195 sge0tsms 42204 subsubmgm 43546 amgmlemALT 44384 |
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