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Theorem mptresid 6008
Description: The restricted identity relation expressed in maps-to notation. (Contributed by FL, 25-Apr-2012.)
Assertion
Ref Expression
mptresid ( I ↾ 𝐴) = (𝑥𝐴𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem mptresid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 opabresid 6007 . 2 ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
2 df-mpt 5193 . 2 (𝑥𝐴𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
31, 2eqtr4i 2764 1 ( I ↾ 𝐴) = (𝑥𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {copab 5171  cmpt 5192   I cid 5534  cres 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-res 5649
This theorem is referenced by:  idref  7096  2fvcoidd  7247  pwfseqlem5  10607  restid2  17320  curf2ndf  18144  hofcl  18156  yonedainv  18178  smndex2dlinvh  18735  sylow1lem2  19389  sylow3lem1  19417  0frgp  19569  frgpcyg  21003  evpmodpmf1o  21023  cnmptid  23035  txswaphmeolem  23178  idnghm  24130  dvexp  25340  dvmptid  25344  mvth  25379  plyid  25593  coeidp  25647  dgrid  25648  plyremlem  25687  taylply2  25750  wilthlem2  26441  ftalem7  26451  fusgrfis  28327  fzto1st1  32007  cycpm2tr  32024  zrhre  32664  qqhre  32665  fsovcnvlem  42377  fourierdlem60  44497  fourierdlem61  44498  itcoval0mpt  46842
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