Theorem ressid 17155

Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Hypothesis

Ref	Expression
ressid.1	⊢ 𝐵 = (Base‘𝑊)

Assertion

Ref	Expression
ressid	⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊)

Proof of Theorem ressid

Step	Hyp	Ref	Expression
1		ssid 3958	. 2 ⊢ 𝐵 ⊆ 𝐵
2		ressid.1	. . 3 ⊢ 𝐵 = (Base‘𝑊)
3	2	fvexi 6836	. 2 ⊢ 𝐵 ∈ V
4		eqid 2729	. . 3 ⊢ (𝑊 ↾s 𝐵) = (𝑊 ↾s 𝐵)
5	4, 2	ressid2 17145	. 2 ⊢ ((𝐵 ⊆ 𝐵 ∧ 𝑊 ∈ 𝑋 ∧ 𝐵 ∈ V) → (𝑊 ↾s 𝐵) = 𝑊)
6	1, 3, 5	mp3an13 1454	1 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ⊆ wss 3903  ‘cfv 6482  (class class class)co 7349  Basecbs 17120   ↾_s cress 17141

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-ress 17142

This theorem is referenced by:  ressval3d  17157  submgmid  18580  submid  18684  subgid  19007  gaid2  19182  subrngid  20434  subrgid  20458  sdrgid  20677  rlmval2  21096  rlmsca  21102  rlmsca2  21103  pjff  21619  dsmmfi  21645  frlmip  21685  evlrhm  22001  evlsscasrng  22002  evlsvarsrng  22004  evl1sca  22219  evl1var  22221  evls1scasrng  22224  evls1varsrng  22225  pf1ind  22240  evl1gsumadd  22243  evl1varpw  22246  ressply1evl  22255  cnstrcvs  25039  cncvs  25043  rlmbn  25259  ishl2  25268  rrxprds  25287  dchrptlem2  27174  evl1fpws  33499  resssra  33553  rgmoddimOLD  33577  qusdimsum  33595  fldextid  33626  riccrng1  42498  ricdrng1  42505  evlsevl  42548  evlvvval  42550  evlvvvallem  42551  mhphf4  42577  lnmfg  43059  lmhmfgsplit  43063  pwslnmlem2  43070  simpcntrab  46855