Theorem resss 4967

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 4672	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss1 3380	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴
3	1, 2	eqsstri 3212	1 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴

Syntax hints:  Vcvv 2760   ∩ cin 3153   ⊆ wss 3154   × cxp 4658   ↾ cres 4662

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167  df-res 4672

This theorem is referenced by:  relssres  4981  resexg  4983  iss  4989  cocnvres  5191  relresfld  5196  relcoi1  5198  funres  5296  funres11  5327  funcnvres  5328  2elresin  5366  fssres  5430  foimacnv  5519  tposss  6301  dftpos4  6318  smores  6347  smores2  6349  caserel  7148  txss12  14445  txbasval  14446