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Theorem dfhe2 36381
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 36380 . 2  |-  ( R hereditary  A 
<->  ( R " A
)  C_  A )
2 rp-imass 36378 . 2  |-  ( ( R " A ) 
C_  A  <->  ( R  |`  A )  C_  ( A  X.  A ) )
31, 2bitri 253 1  |-  ( R hereditary  A 
<->  ( R  |`  A ) 
C_  ( A  X.  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    C_ wss 3406    X. cxp 4835    |` cres 4839   "cima 4840   hereditary whe 36379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-he 36380
This theorem is referenced by:  xpheOLD  36389  idhe  36395
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