Theorem mptresid 6016

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 6015	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5167	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2762	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {copab 5147   ↦ cmpt 5166   I cid 5525   ↾ cres 5633

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-res 5643

This theorem is referenced by:  idref  7099  2fvcoidd  7252  pwfseqlem5  10586  restid2  17393  curf2ndf  18213  hofcl  18225  yonedainv  18247  smndex2dlinvh  18888  sylow1lem2  19574  sylow3lem1  19602  0frgp  19754  frgpcyg  21553  evpmodpmf1o  21576  cnmptid  23626  txswaphmeolem  23769  idnghm  24708  dvexp  25920  dvmptid  25924  mvth  25959  plyid  26174  coeidp  26228  dgrid  26229  plyremlem  26270  taylply2  26333  wilthlem2  27032  ftalem7  27042  fusgrfis  29399  fzto1st1  33163  cycpm2tr  33180  zrhre  34163  qqhre  34164  fsovcnvlem  44440  fourierdlem60  46594  fourierdlem61  46595  itcoval0mpt  49142