Theorem mptresid 5892

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 5891	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5111	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2764	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 399   = wceq 1542   ∈ wcel 2114  {copab 5092   ↦ cmpt 5110   I cid 5428   ↾ cres 5527

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-res 5537

This theorem is referenced by:  idref  6918  2fvcoidd  7064  pwfseqlem5  10163  restid2  16807  curf2ndf  17613  hofcl  17625  yonedainv  17647  smndex2dlinvh  18198  sylow1lem2  18842  sylow3lem1  18870  0frgp  19023  frgpcyg  20392  evpmodpmf1o  20412  cnmptid  22412  txswaphmeolem  22555  idnghm  23496  dvexp  24705  dvmptid  24709  mvth  24744  plyid  24958  coeidp  25012  dgrid  25013  plyremlem  25052  taylply2  25115  wilthlem2  25806  ftalem7  25816  fusgrfis  27272  fzto1st1  30946  cycpm2tr  30963  zrhre  31539  qqhre  31540  fsovcnvlem  41167  fourierdlem60  43249  fourierdlem61  43250  itcoval0mpt  45546