Theorem mptresid 6010

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 6009	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5168	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2763	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {copab 5148   ↦ cmpt 5167   I cid 5518   ↾ cres 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-res 5636

This theorem is referenced by:  idref  7093  2fvcoidd  7245  pwfseqlem5  10577  restid2  17384  curf2ndf  18204  hofcl  18216  yonedainv  18238  smndex2dlinvh  18879  sylow1lem2  19565  sylow3lem1  19593  0frgp  19745  frgpcyg  21563  evpmodpmf1o  21586  cnmptid  23636  txswaphmeolem  23779  idnghm  24718  dvexp  25930  dvmptid  25934  mvth  25969  plyid  26184  coeidp  26238  dgrid  26239  plyremlem  26281  taylply2  26344  taylply2OLD  26345  wilthlem2  27046  ftalem7  27056  fusgrfis  29413  fzto1st1  33178  cycpm2tr  33195  zrhre  34179  qqhre  34180  fsovcnvlem  44458  fourierdlem60  46612  fourierdlem61  46613  itcoval0mpt  49154