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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 3986 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
2 | ressid.1 | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | fvexi 6677 | . 2 ⊢ 𝐵 ∈ V |
4 | eqid 2818 | . . 3 ⊢ (𝑊 ↾s 𝐵) = (𝑊 ↾s 𝐵) | |
5 | 4, 2 | ressid2 16540 | . 2 ⊢ ((𝐵 ⊆ 𝐵 ∧ 𝑊 ∈ 𝑋 ∧ 𝐵 ∈ V) → (𝑊 ↾s 𝐵) = 𝑊) |
6 | 1, 3, 5 | mp3an13 1443 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 ↾s cress 16472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-ress 16479 |
This theorem is referenced by: ressval3d 16549 submid 17963 subgid 18219 gaid2 18371 subrgid 19466 sdrgid 19504 rlmval2 19895 rlmsca 19901 rlmsca2 19902 evlrhm 20237 evlsscasrng 20238 evlsvarsrng 20240 evl1sca 20425 evl1var 20427 evls1scasrng 20430 evls1varsrng 20431 pf1ind 20446 evl1gsumadd 20449 evl1varpw 20452 pjff 20784 dsmmfi 20810 frlmip 20850 cnstrcvs 23672 cncvs 23676 rlmbn 23891 ishl2 23900 rrxprds 23919 dchrptlem2 25768 rgmoddim 30907 qusdimsum 30923 fldextid 30948 lnmfg 39560 lmhmfgsplit 39564 pwslnmlem2 39571 simpcntrab 43004 submgmid 43937 |
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