Theorem resid 4923

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 3150	. 2 |- dom A C_ _V
2		relssres 4905	. 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 1335   _Vcvv 2712   C_ wss 3102   dom cdm 4587   |` cres 4589   Rel wrel 4592

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-rel 4594  df-dm 4597  df-res 4599

This theorem is referenced by: (None)