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Theorem resss 4578
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 4300 . 2 (AB) = (A ∩ (B × V))
2 inss1 3151 . 2 (A ∩ (B × V)) ⊆ A
31, 2eqsstri 2969 1 (AB) ⊆ A
Colors of variables: wff set class
Syntax hints:  Vcvv 2551  cin 2910  wss 2911   × cxp 4286  cres 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-res 4300
This theorem is referenced by:  relssres  4591  resexg  4593  iss  4597  relresfld  4790  relcoi1  4792  funres  4884  funres11  4914  funcnvres  4915  2elresin  4953  fssres  5009  foimacnv  5087  tposss  5802  dftpos4  5819  smores  5848  smores2  5850
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