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Theorem dfhe2 38307
Description: The property of relation  R being hereditary in class  A. (Contributed by RP, 27-Mar-2020.)
Assertion
Ref Expression
dfhe2  |-  ( R hereditary  A 
<->  ( R  |`  A ) 
C_  ( A  X.  A ) )

Proof of Theorem dfhe2
StepHypRef Expression
1 df-he 38306 . 2  |-  ( R hereditary  A 
<->  ( R " A
)  C_  A )
2 rp-imass 38304 . 2  |-  ( ( R " A ) 
C_  A  <->  ( R  |`  A )  C_  ( A  X.  A ) )
31, 2bitri 249 1  |-  ( R hereditary  A 
<->  ( R  |`  A ) 
C_  ( A  X.  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    C_ wss 3406    X. cxp 4928    |` cres 4932   "cima 4933   hereditary whe 38305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-br 4385  df-opab 4443  df-xp 4936  df-rel 4937  df-cnv 4938  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-he 38306
This theorem is referenced by:  xpheOLD  38315  idhe  38321
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