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Mirrors > Home > MPE Home > Th. List > ressid | Structured version Visualization version GIF version |
Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
Ref | Expression |
---|---|
ressid.1 | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
ressid | ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 4001 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
2 | ressid.1 | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | fvexi 6893 | . 2 ⊢ 𝐵 ∈ V |
4 | eqid 2732 | . . 3 ⊢ (𝑊 ↾s 𝐵) = (𝑊 ↾s 𝐵) | |
5 | 4, 2 | ressid2 17161 | . 2 ⊢ ((𝐵 ⊆ 𝐵 ∧ 𝑊 ∈ 𝑋 ∧ 𝐵 ∈ V) → (𝑊 ↾s 𝐵) = 𝑊) |
6 | 1, 3, 5 | mp3an13 1452 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3945 ‘cfv 6533 (class class class)co 7394 Basecbs 17128 ↾s cress 17157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-ress 17158 |
This theorem is referenced by: ressval3d 17175 ressval3dOLD 17176 submid 18669 subgid 18982 gaid2 19135 subrgid 20316 sdrgid 20359 rlmval2 20766 rlmsca 20772 rlmsca2 20773 pjff 21202 dsmmfi 21228 frlmip 21268 evlrhm 21590 evlsscasrng 21591 evlsvarsrng 21593 evl1sca 21784 evl1var 21786 evls1scasrng 21789 evls1varsrng 21790 pf1ind 21805 evl1gsumadd 21808 evl1varpw 21811 cnstrcvs 24588 cncvs 24592 rlmbn 24809 ishl2 24818 rrxprds 24837 dchrptlem2 26697 ressply1evl 32554 rgmoddim 32597 qusdimsum 32615 fldextid 32640 riccrng1 40964 ricdrng1 40970 evlsevl 41002 mhphf4 41024 lnmfg 41659 lmhmfgsplit 41663 pwslnmlem2 41670 simpcntrab 45423 submgmid 46399 |
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