Theorem resss 4988

Description: A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 2-Aug-1994.)

Assertion

Ref	Expression
resss	⊢ (A ↾ B) ⊆ A

Proof of Theorem resss

Step	Hyp	Ref	Expression
1		df-res 4788	. 2 ⊢ (A ↾ B) = (A ∩ (B × V))
2		inss1 3475	. 2 ⊢ (A ∩ (B × V)) ⊆ A
3	1, 2	eqsstri 3301	1 ⊢ (A ↾ B) ⊆ A

Syntax hints:  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257   × cxp 4770   ↾ cres 4774

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-res 4788

This theorem is referenced by:  ssreseq  4997  iss  5000  funres  5143  funres11  5164  funcnvres  5165  2elresin  5194  fssres  5238  foimacnv  5303