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Theorem resss 4811
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 4519 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 inss1 3264 . 2 (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴
31, 2eqsstri 3097 1 (𝐴𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  Vcvv 2658  cin 3038  wss 3039   × cxp 4505  cres 4509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-res 4519
This theorem is referenced by:  relssres  4825  resexg  4827  iss  4833  cocnvres  5031  relresfld  5036  relcoi1  5038  funres  5132  funres11  5163  funcnvres  5164  2elresin  5202  fssres  5266  foimacnv  5351  tposss  6109  dftpos4  6126  smores  6155  smores2  6157  caserel  6938  txss12  12330  txbasval  12331
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