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 5161	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2766	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {copab 5141   ↦ cmpt 5160   I cid 5519   ↾ cres 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-res 5637

This theorem is referenced by:  idref  7095  2fvcoidd  7248  pwfseqlem5  10584  restid2  17391  curf2ndf  18211  hofcl  18223  yonedainv  18245  smndex2dlinvh  18886  sylow1lem2  19572  sylow3lem1  19600  0frgp  19752  frgpcyg  21555  evpmodpmf1o  21578  cnmptid  23651  txswaphmeolem  23794  idnghm  24733  dvexp  25945  dvmptid  25949  mvth  25984  plyid  26199  coeidp  26253  dgrid  26254  plyremlem  26295  taylply2  26358  wilthlem2  27057  ftalem7  27067  fusgrfis  29424  fzto1st1  33190  cycpm2tr  33207  zrhre  34210  qqhre  34211  fsovcnvlem  44464  fourierdlem60  46616  fourierdlem61  46617  itcoval0mpt  49164