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

Step	Hyp	Ref	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 ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( _I |` A ) C_ ( dom ( _I |` A ) X. ran ( _I |` A ) )
4		dmresi 5160	. . . . 5 |- dom ( _I |` A ) = A
5	4	eqimssi 3486	. . . 4 |- dom ( _I |` A ) C_ A
6		rnresi 5181	. . . . 5 |- ran ( _I |` A ) = A
7	6	eqimssi 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 ) )
9	5, 7, 8	mp2an 678	. . 3 |- ( dom ( _I |` A ) X. ran ( _I |` A ) ) C_ ( A X. A )
10	3, 9	sstri 3441	. 2 |- ( _I |` A ) C_ ( A X. A )
11		dfhe2 36369	. 2 |- ( _I hereditary A <-> ( _I |` A ) C_ ( A X. A ) )
12	10, 11	mpbir 213	1 |- _I hereditary A

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