Theorem mptresid 6055

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 6054	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5190	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2789	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {copab 5170   ↦ cmpt 5189   I cid 5549   ↾ cres 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-res 5667

This theorem is used by:  idref  7173  2fvcoidd  7328  pwfseqlem5  10775  restid2  17630  curf2ndf  18450  hofcl  18462  yonedainv  18484  smndex2dlinvh  19145  sylow1lem2  19843  sylow3lem1  19871  0frgp  20023  frgpcyg  21909  evpmodpmf1o  21932  cnmptid  24010  txswaphmeolem  24153  idnghm  25092  dvexp  26303  dvmptid  26307  mvth  26342  plyid  26557  coeidp  26612  dgrid  26613  plyremlem  26655  taylply2  26725  wilthlem2  27426  ftalem7  27436  fusgrfis  29941  fzto1st1  33733  cycpm2tr  33750  zrhre  34721  qqhre  34722  fsovcnvlem  45205  fourierdlem60  47381  fourierdlem61  47382  itcoval0mpt  49983  termcpostfix  50877  fuconuniqlem  50986  fuconuniqvalslem  50988