Theorem resdm 5097

Description: A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)

Assertion

Ref	Expression
resdm	⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)

Proof of Theorem resdm

Step	Hyp	Ref	Expression
1		ssid 3268	. 2 ⊢ dom 𝐴 ⊆ dom 𝐴
2		relssres 5096	. 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴)
3	1, 2	mpan2 429	1 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1402   ⊆ wss 3220  dom cdm 4769   ↾ cres 4771  Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-dm 4779  df-res 4781

This theorem is referenced by:  resindm  5100  resdm2  5273  relresfld  5312  relcoi1  5314  funimaexg  5460  fnex  5928  dftpos2  6522  pmresg  6947  dif1en  7173