Theorem resss 4724

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 4440	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss1 3218	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴
3	1, 2	eqsstri 3054	1 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴

Syntax hints:  Vcvv 2619   ∩ cin 2996   ⊆ wss 2997   × cxp 4426   ↾ cres 4430

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-res 4440

This theorem is referenced by:  relssres  4737  resexg  4739  iss  4745  cocnvres  4942  relresfld  4947  relcoi1  4949  funres  5041  funres11  5072  funcnvres  5073  2elresin  5111  fssres  5171  foimacnv  5255  tposss  5993  dftpos4  6010  smores  6039  smores2  6041  caserel  6757