Theorem resss 4838

Description: A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Assertion

Ref	Expression
resss	⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴

Proof of Theorem resss

Step	Hyp	Ref	Expression
1		df-res 4546	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss1 3291	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴
3	1, 2	eqsstri 3124	1 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴

Syntax hints:  Vcvv 2681   ∩ cin 3065   ⊆ wss 3066   × cxp 4532   ↾ cres 4536

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-res 4546

This theorem is referenced by:  relssres  4852  resexg  4854  iss  4860  cocnvres  5058  relresfld  5063  relcoi1  5065  funres  5159  funres11  5190  funcnvres  5191  2elresin  5229  fssres  5293  foimacnv  5378  tposss  6136  dftpos4  6153  smores  6182  smores2  6184  caserel  6965  txss12  12424  txbasval  12425