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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 4006 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
| 2 | ressid.1 | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
| 3 | 2 | fvexi 6920 | . 2 ⊢ 𝐵 ∈ V |
| 4 | eqid 2737 | . . 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 1454 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-ress 17275 |
| This theorem is referenced by: ressval3d 17292 submgmid 18719 submid 18823 subgid 19146 gaid2 19321 subrngid 20549 subrgid 20573 sdrgid 20793 rlmval2 21199 rlmsca 21205 rlmsca2 21206 pjff 21732 dsmmfi 21758 frlmip 21798 evlrhm 22120 evlsscasrng 22121 evlsvarsrng 22123 evl1sca 22338 evl1var 22340 evls1scasrng 22343 evls1varsrng 22344 pf1ind 22359 evl1gsumadd 22362 evl1varpw 22365 ressply1evl 22374 cnstrcvs 25174 cncvs 25178 rlmbn 25395 ishl2 25404 rrxprds 25423 dchrptlem2 27309 evl1fpws 33590 resssra 33638 rgmoddimOLD 33661 qusdimsum 33679 fldextid 33710 riccrng1 42531 ricdrng1 42538 evlsevl 42581 evlvvval 42583 evlvvvallem 42584 mhphf4 42610 lnmfg 43094 lmhmfgsplit 43098 pwslnmlem2 43105 simpcntrab 46885 |
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