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Theorem resss 4932
Description: A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
Assertion
Ref Expression
resss (𝐴𝐵) ⊆ 𝐴

Proof of Theorem resss
StepHypRef Expression
1 df-res 4639 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 inss1 3356 . 2 (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴
31, 2eqsstri 3188 1 (𝐴𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  Vcvv 2738  cin 3129  wss 3130   × cxp 4625  cres 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-res 4639
This theorem is referenced by:  relssres  4946  resexg  4948  iss  4954  cocnvres  5154  relresfld  5159  relcoi1  5161  funres  5258  funres11  5289  funcnvres  5290  2elresin  5328  fssres  5392  foimacnv  5480  tposss  6247  dftpos4  6264  smores  6293  smores2  6295  caserel  7086  txss12  13769  txbasval  13770
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