Theorem resss 4843

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 4551	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss1 3296	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴
3	1, 2	eqsstri 3129	1 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴

Syntax hints:  Vcvv 2686   ∩ cin 3070   ⊆ wss 3071   × cxp 4537   ↾ cres 4541

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-res 4551

This theorem is referenced by:  relssres  4857  resexg  4859  iss  4865  cocnvres  5063  relresfld  5068  relcoi1  5070  funres  5164  funres11  5195  funcnvres  5196  2elresin  5234  fssres  5298  foimacnv  5385  tposss  6143  dftpos4  6160  smores  6189  smores2  6191  caserel  6972  txss12  12435  txbasval  12436