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

Step	Hyp	Ref	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 ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( _I |` A ) C_ ( dom ( _I |` A ) X. ran ( _I |` A ) )
4		dmresi 5166	. . . . 5 |- dom ( _I |` A ) = A
5	4	eqimssi 3472	. . . 4 |- dom ( _I |` A ) C_ A
6		rnresi 5187	. . . . 5 |- ran ( _I |` A ) = A
7	6	eqimssi 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 ) )
9	5, 7, 8	mp2an 686	. . 3 |- ( dom ( _I |` A ) X. ran ( _I |` A ) ) C_ ( A X. A )
10	3, 9	sstri 3427	. 2 |- ( _I |` A ) C_ ( A X. A )
11		dfhe2 36440	. 2 |- ( _I hereditary A <-> ( _I |` A ) C_ ( A X. A ) )
12	10, 11	mpbir 214	1 |- _I hereditary A

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