Theorem mptresid 6051

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 6050	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5233	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2764	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {copab 5211   ↦ cmpt 5232   I cid 5574   ↾ cres 5679

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 5300  ax-nul 5307  ax-pr 5428

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 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-res 5689

This theorem is referenced by:  idref  7144  2fvcoidd  7295  pwfseqlem5  10658  restid2  17376  curf2ndf  18200  hofcl  18212  yonedainv  18234  smndex2dlinvh  18798  sylow1lem2  19467  sylow3lem1  19495  0frgp  19647  frgpcyg  21129  evpmodpmf1o  21149  cnmptid  23165  txswaphmeolem  23308  idnghm  24260  dvexp  25470  dvmptid  25474  mvth  25509  plyid  25723  coeidp  25777  dgrid  25778  plyremlem  25817  taylply2  25880  wilthlem2  26573  ftalem7  26583  fusgrfis  28587  fzto1st1  32261  cycpm2tr  32278  zrhre  32999  qqhre  33000  fsovcnvlem  42764  fourierdlem60  44882  fourierdlem61  44883  itcoval0mpt  47352