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Theorem mptresid 5444
Description: The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
Assertion
Ref Expression
mptresid (𝑥𝐴𝑥) = ( I ↾ 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem mptresid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4721 . 2 (𝑥𝐴𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
2 opabresid 5443 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
31, 2eqtri 2642 1 (𝑥𝐴𝑥) = ( I ↾ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1481  wcel 1988  {copab 4703  cmpt 4720   I cid 5013  cres 5106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-res 5116
This theorem is referenced by:  idref  6484  2fvcoidd  6537  pwfseqlem5  9470  restid2  16072  curf2ndf  16868  hofcl  16880  yonedainv  16902  sylow1lem2  17995  sylow3lem1  18023  0frgp  18173  frgpcyg  19903  evpmodpmf1o  19923  txswaphmeolem  21588  idnghm  22528  dvexp  23697  dvmptid  23701  mvth  23736  plyid  23946  coeidp  24000  dgrid  24001  plyremlem  24040  taylply2  24103  wilthlem2  24776  ftalem7  24786  fusgrfis  26203  fzto1st1  29826  zrhre  30037  qqhre  30038  fsovcnvlem  38127  fourierdlem60  40146  fourierdlem61  40147
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