Theorem mptresid 5035

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 5034	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 4126	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2233	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 104   = wceq 1375   ∈ wcel 2180  {copab 4123   ↦ cmpt 4124   I cid 4356   ↾ cres 4698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-res 4708

This theorem is referenced by:  idref  5853  restid2  13247  txswaphmeolem  14959  dvexp  15350  dvmptidcn  15353  dvmptid  15355  plyid  15385