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 5180	. 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 1541   ∈ wcel 2113  {copab 5160   ↦ cmpt 5179   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 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-res 5636

This theorem is referenced by:  idref  7091  2fvcoidd  7243  pwfseqlem5  10574  restid2  17350  curf2ndf  18170  hofcl  18182  yonedainv  18204  smndex2dlinvh  18842  sylow1lem2  19528  sylow3lem1  19556  0frgp  19708  frgpcyg  21528  evpmodpmf1o  21551  cnmptid  23605  txswaphmeolem  23748  idnghm  24687  dvexp  25913  dvmptid  25917  mvth  25953  plyid  26170  coeidp  26225  dgrid  26226  plyremlem  26268  taylply2  26331  taylply2OLD  26332  wilthlem2  27035  ftalem7  27045  fusgrfis  29403  fzto1st1  33184  cycpm2tr  33201  zrhre  34176  qqhre  34177  fsovcnvlem  44250  fourierdlem60  46406  fourierdlem61  46407  itcoval0mpt  48908