Theorem resss 5087

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 4786	. 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 4772   ↾ cres 4776

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 4786

This theorem is used by:  relssres  5101  resexg  5103  iss  5109  cocnvres  5312  relresfld  5317  relcoi1  5319  funres  5418  funres11  5453  funcnvres  5454  2elresin  5494  fssres  5565  foimacnv  5657  tposss  6517  dftpos4  6534  smores  6563  smores2  6565  caserel  7427  txss12  15367  txbasval  15368  issubgr2  16499  subgrprop2  16501  uhgrspansubgr  16518