Theorem mptresid 6050

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 6049	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5232	. 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 1540   ∈ wcel 2105  {copab 5210   ↦ cmpt 5231   I cid 5573   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-res 5688

This theorem is referenced by:  idref  7146  2fvcoidd  7298  pwfseqlem5  10664  restid2  17383  curf2ndf  18207  hofcl  18219  yonedainv  18241  smndex2dlinvh  18837  sylow1lem2  19512  sylow3lem1  19540  0frgp  19692  frgpcyg  21352  evpmodpmf1o  21372  cnmptid  23398  txswaphmeolem  23541  idnghm  24493  dvexp  25718  dvmptid  25722  mvth  25758  plyid  25972  coeidp  26027  dgrid  26028  plyremlem  26067  taylply2  26130  wilthlem2  26824  ftalem7  26834  fusgrfis  28869  fzto1st1  32546  cycpm2tr  32563  zrhre  33312  qqhre  33313  fsovcnvlem  43079  fourierdlem60  45193  fourierdlem61  45194  itcoval0mpt  47452