Theorem mptresid 6022

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 6021	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5189	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2755	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {copab 5169   ↦ cmpt 5188   I cid 5532   ↾ cres 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-res 5650

This theorem is referenced by:  idref  7118  2fvcoidd  7272  pwfseqlem5  10616  restid2  17393  curf2ndf  18208  hofcl  18220  yonedainv  18242  smndex2dlinvh  18844  sylow1lem2  19529  sylow3lem1  19557  0frgp  19709  frgpcyg  21483  evpmodpmf1o  21505  cnmptid  23548  txswaphmeolem  23691  idnghm  24631  dvexp  25857  dvmptid  25861  mvth  25897  plyid  26114  coeidp  26169  dgrid  26170  plyremlem  26212  taylply2  26275  taylply2OLD  26276  wilthlem2  26979  ftalem7  26989  fusgrfis  29257  fzto1st1  33059  cycpm2tr  33076  zrhre  34009  qqhre  34010  fsovcnvlem  44002  fourierdlem60  46164  fourierdlem61  46165  itcoval0mpt  48655