Theorem mptresid 6058

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 6057	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5198	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2792	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {copab 5178   ↦ cmpt 5197   I cid 5560   ↾ cres 5668

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-res 5678

This theorem is used by:  idref  7149  2fvcoidd  7306  pwfseqlem5  10666  restid2  17508  curf2ndf  18328  hofcl  18340  yonedainv  18362  smndex2dlinvh  19010  sylow1lem2  19700  sylow3lem1  19728  0frgp  19880  frgpcyg  21760  evpmodpmf1o  21783  cnmptid  23855  txswaphmeolem  23998  idnghm  24937  dvexp  26149  dvmptid  26153  mvth  26188  plyid  26403  coeidp  26457  dgrid  26458  plyremlem  26502  taylply2  26568  wilthlem2  27270  ftalem7  27280  fusgrfis  29717  fzto1st1  33453  cycpm2tr  33470  zrhre  34440  qqhre  34441  fsovcnvlem  44780  fourierdlem60  46921  fourierdlem61  46922  itcoval0mpt  49487