Theorem mptresid 6037

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 6036	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5181	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2787	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {copab 5161   ↦ cmpt 5180   I cid 5539   ↾ cres 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-res 5657

This theorem is referenced by:  idref  7124  2fvcoidd  7277  pwfseqlem5  10618  restid2  17442  curf2ndf  18262  hofcl  18274  yonedainv  18296  smndex2dlinvh  18937  sylow1lem2  19622  sylow3lem1  19650  0frgp  19802  frgpcyg  21605  evpmodpmf1o  21628  cnmptid  23701  txswaphmeolem  23844  idnghm  24783  dvexp  25995  dvmptid  25999  mvth  26034  plyid  26249  coeidp  26303  dgrid  26304  plyremlem  26345  taylply2  26408  wilthlem2  27110  ftalem7  27120  fusgrfis  29477  fzto1st1  33243  cycpm2tr  33260  zrhre  34277  qqhre  34278  fsovcnvlem  44553  fourierdlem60  46704  fourierdlem61  46705  itcoval0mpt  49252