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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 4018 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
2 | ressid.1 | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | fvexi 6921 | . 2 ⊢ 𝐵 ∈ V |
4 | eqid 2735 | . . 3 ⊢ (𝑊 ↾s 𝐵) = (𝑊 ↾s 𝐵) | |
5 | 4, 2 | ressid2 17278 | . 2 ⊢ ((𝐵 ⊆ 𝐵 ∧ 𝑊 ∈ 𝑋 ∧ 𝐵 ∈ V) → (𝑊 ↾s 𝐵) = 𝑊) |
6 | 1, 3, 5 | mp3an13 1451 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-ress 17275 |
This theorem is referenced by: ressval3d 17292 ressval3dOLD 17293 submgmid 18732 submid 18836 subgid 19159 gaid2 19334 subrngid 20566 subrgid 20590 sdrgid 20810 rlmval2 21217 rlmsca 21223 rlmsca2 21224 pjff 21750 dsmmfi 21776 frlmip 21816 evlrhm 22138 evlsscasrng 22139 evlsvarsrng 22141 evl1sca 22354 evl1var 22356 evls1scasrng 22359 evls1varsrng 22360 pf1ind 22375 evl1gsumadd 22378 evl1varpw 22381 ressply1evl 22390 cnstrcvs 25188 cncvs 25192 rlmbn 25409 ishl2 25418 rrxprds 25437 dchrptlem2 27324 evl1fpws 33570 resssra 33617 rgmoddimOLD 33638 qusdimsum 33656 fldextid 33687 riccrng1 42508 ricdrng1 42515 evlsevl 42558 evlvvval 42560 evlvvvallem 42561 mhphf4 42587 lnmfg 43071 lmhmfgsplit 43075 pwslnmlem2 43082 simpcntrab 46826 |
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