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Theorem idhe 36383
Description: The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
Assertion
Ref Expression
idhe  |-  _I hereditary  A

Proof of Theorem idhe
StepHypRef Expression
1 relres 5132 . . . 4  |-  Rel  (  _I  |`  A )
2 relssdmrn 5356 . . . 4  |-  ( Rel  (  _I  |`  A )  ->  (  _I  |`  A ) 
C_  ( dom  (  _I  |`  A )  X. 
ran  (  _I  |`  A ) ) )
31, 2ax-mp 5 . . 3  |-  (  _I  |`  A )  C_  ( dom  (  _I  |`  A )  X.  ran  (  _I  |`  A ) )
4 dmresi 5160 . . . . 5  |-  dom  (  _I  |`  A )  =  A
54eqimssi 3486 . . . 4  |-  dom  (  _I  |`  A )  C_  A
6 rnresi 5181 . . . . 5  |-  ran  (  _I  |`  A )  =  A
76eqimssi 3486 . . . 4  |-  ran  (  _I  |`  A )  C_  A
8 xpss12 4940 . . . 4  |-  ( ( dom  (  _I  |`  A ) 
C_  A  /\  ran  (  _I  |`  A ) 
C_  A )  -> 
( dom  (  _I  |`  A )  X.  ran  (  _I  |`  A ) )  C_  ( A  X.  A ) )
95, 7, 8mp2an 678 . . 3  |-  ( dom  (  _I  |`  A )  X.  ran  (  _I  |`  A ) )  C_  ( A  X.  A
)
103, 9sstri 3441 . 2  |-  (  _I  |`  A )  C_  ( A  X.  A )
11 dfhe2 36369 . 2  |-  (  _I hereditary  A  <-> 
(  _I  |`  A ) 
C_  ( A  X.  A ) )
1210, 11mpbir 213 1  |-  _I hereditary  A
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3404    _I cid 4744    X. cxp 4832   dom cdm 4834   ran crn 4835    |` cres 4836   Rel wrel 4839   hereditary whe 36367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-he 36368
This theorem is referenced by:  sshepw  36385
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