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Theorem idhe 36454
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 5138 . . . 4  |-  Rel  (  _I  |`  A )
2 relssdmrn 5363 . . . 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 5166 . . . . 5  |-  dom  (  _I  |`  A )  =  A
54eqimssi 3472 . . . 4  |-  dom  (  _I  |`  A )  C_  A
6 rnresi 5187 . . . . 5  |-  ran  (  _I  |`  A )  =  A
76eqimssi 3472 . . . 4  |-  ran  (  _I  |`  A )  C_  A
8 xpss12 4945 . . . 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 686 . . 3  |-  ( dom  (  _I  |`  A )  X.  ran  (  _I  |`  A ) )  C_  ( A  X.  A
)
103, 9sstri 3427 . 2  |-  (  _I  |`  A )  C_  ( A  X.  A )
11 dfhe2 36440 . 2  |-  (  _I hereditary  A  <-> 
(  _I  |`  A ) 
C_  ( A  X.  A ) )
1210, 11mpbir 214 1  |-  _I hereditary  A
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3390    _I cid 4749    X. cxp 4837   dom cdm 4839   ran crn 4840    |` cres 4841   Rel wrel 4844   hereditary whe 36438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-he 36439
This theorem is referenced by:  sshepw  36456
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