Theorem resid 4939

Description: Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)

Assertion

Ref	Expression
resid	|- ( Rel A -> ( A |` _V ) = A )

Proof of Theorem resid

Step	Hyp	Ref	Expression
1		ssv 3163	. 2 |- dom A C_ _V
2		relssres 4921	. 2 |- ( ( Rel A /\ dom A C_ _V ) -> ( A |` _V ) = A )
3	1, 2	mpan2 422	1 |- ( Rel A -> ( A |` _V ) = A )

Syntax hints:   -> wi 4   = wceq 1343   _Vcvv 2725   C_ wss 3115   dom cdm 4603   |` cres 4605   Rel wrel 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-br 3982  df-opab 4043  df-xp 4609  df-rel 4610  df-dm 4613  df-res 4615

This theorem is referenced by: (None)