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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 3587 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
2 | ressid.1 | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | fvex 6113 | . . 3 ⊢ (Base‘𝑊) ∈ V | |
4 | 2, 3 | eqeltri 2684 | . 2 ⊢ 𝐵 ∈ V |
5 | eqid 2610 | . . 3 ⊢ (𝑊 ↾s 𝐵) = (𝑊 ↾s 𝐵) | |
6 | 5, 2 | ressid2 15755 | . 2 ⊢ ((𝐵 ⊆ 𝐵 ∧ 𝑊 ∈ 𝑋 ∧ 𝐵 ∈ V) → (𝑊 ↾s 𝐵) = 𝑊) |
7 | 1, 4, 6 | mp3an13 1407 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-ress 15702 |
This theorem is referenced by: ressval3d 15764 submid 17174 subgid 17419 gaid2 17559 subrgid 18605 rlmval2 19015 rlmsca 19021 rlmsca2 19022 evlrhm 19346 evlsscasrng 19347 evlsvarsrng 19349 evl1sca 19519 evl1var 19521 evls1scasrng 19524 evls1varsrng 19525 pf1ind 19540 evl1gsumadd 19543 evl1varpw 19546 pjff 19875 dsmmfi 19901 frlmip 19936 cnstrcvs 22749 cncvs 22753 rlmbn 22965 ishl2 22974 rrxprds 22985 dchrptlem2 24790 lnmfg 36670 lmhmfgsplit 36674 pwslnmlem2 36681 submgmid 41583 |
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