Theorem resss 4989

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 4789	. 2 ⊢ (A ↾ B) = (A ∩ (B × V))
2		inss1 3476	. 2 ⊢ (A ∩ (B × V)) ⊆ A
3	1, 2	eqsstri 3302	1 ⊢ (A ↾ B) ⊆ A

Syntax hints:  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258   × cxp 4771   ↾ cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-res 4789

This theorem is referenced by:  ssreseq  4998  iss  5001  funres  5144  funres11  5165  funcnvres  5166  2elresin  5195  fssres  5239  foimacnv  5304