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Theorem mptresid 5359
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 4636 . 2 (𝑥𝐴𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
2 opabresid 5358 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
31, 2eqtri 2628 1 (𝑥𝐴𝑥) = ( I ↾ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wcel 1976  {copab 4633  cmpt 4634   I cid 4935  cres 5027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-res 5037
This theorem is referenced by:  idref  6378  2fvcoidd  6427  pwfseqlem5  9338  restid2  15857  curf2ndf  16653  hofcl  16665  yonedainv  16687  sylow1lem2  17780  sylow3lem1  17808  0frgp  17958  frgpcyg  19683  evpmodpmf1o  19703  txswaphmeolem  21356  idnghm  22286  dvexp  23436  dvmptid  23440  mvth  23473  plyid  23683  coeidp  23737  dgrid  23738  plyremlem  23777  taylply2  23840  wilthlem2  24509  ftalem7  24519  fzto1st1  28986  zrhre  29194  qqhre  29195  fsovcnvlem  37127  fourierdlem60  38860  fourierdlem61  38861  fusgrfis  40548
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