Theorem mptresid 6080

Description: The restricted identity relation expressed in maps-to notation. (Contributed by FL, 25-Apr-2012.)

Assertion

Ref	Expression
mptresid	⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem mptresid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabresid 6079	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5250	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2771	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {copab 5228   ↦ cmpt 5249   I cid 5592   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-res 5712

This theorem is referenced by:  idref  7180  2fvcoidd  7333  pwfseqlem5  10732  restid2  17490  curf2ndf  18317  hofcl  18329  yonedainv  18351  smndex2dlinvh  18952  sylow1lem2  19641  sylow3lem1  19669  0frgp  19821  frgpcyg  21615  evpmodpmf1o  21637  cnmptid  23690  txswaphmeolem  23833  idnghm  24785  dvexp  26011  dvmptid  26015  mvth  26051  plyid  26268  coeidp  26323  dgrid  26324  plyremlem  26364  taylply2  26427  taylply2OLD  26428  wilthlem2  27130  ftalem7  27140  fusgrfis  29365  fzto1st1  33095  cycpm2tr  33112  zrhre  33965  qqhre  33966  fsovcnvlem  43975  fourierdlem60  46087  fourierdlem61  46088  itcoval0mpt  48400