Theorem resss 5089

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 4788	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss1 3451	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴
3	1, 2	eqsstri 3280	1 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴

This proof depends on syntax axioms:  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220   × cxp 4774   ↾ cres 4778

This proof depends on 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-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-res 4788

This theorem is used by:  relssres  5103  resexg  5105  iss  5112  cocnvres  5315  relresfld  5320  relcoi1  5322  funres  5421  funres11  5456  funcnvres  5457  2elresin  5497  fssres  5568  foimacnv  5660  tposss  6521  dftpos4  6538  smores  6567  smores2  6569  relresfin  7268  caserel  7433  gsum2dfilem2  14272  txss12  15492  txbasval  15493  issubgr2  16704  subgrprop2  16706  uhgrspansubgr  16723