Theorem resss 5029

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 4731	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss1 3424	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴
3	1, 2	eqsstri 3256	1 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴

Syntax hints:  Vcvv 2799   ∩ cin 3196   ⊆ wss 3197   × cxp 4717   ↾ cres 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-res 4731

This theorem is referenced by:  relssres  5043  resexg  5045  iss  5051  cocnvres  5253  relresfld  5258  relcoi1  5260  funres  5359  funres11  5393  funcnvres  5394  2elresin  5434  fssres  5503  foimacnv  5592  tposss  6398  dftpos4  6415  smores  6444  smores2  6446  caserel  7265  txss12  14955  txbasval  14956