Theorem idhe 38081

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 5426	. . . 4 ⊢ Rel ( I ↾ 𝐴)
2		relssdmrn 5656	. . . 4 ⊢ (Rel ( I ↾ 𝐴) → ( I ↾ 𝐴) ⊆ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)))
3	1, 2	ax-mp 5	. . 3 ⊢ ( I ↾ 𝐴) ⊆ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴))
4		dmresi 5457	. . . . 5 ⊢ dom ( I ↾ 𝐴) = 𝐴
5	4	eqimssi 3659	. . . 4 ⊢ dom ( I ↾ 𝐴) ⊆ 𝐴
6		rnresi 5479	. . . . 5 ⊢ ran ( I ↾ 𝐴) = 𝐴
7	6	eqimssi 3659	. . . 4 ⊢ ran ( I ↾ 𝐴) ⊆ 𝐴
8		xpss12 5225	. . . 4 ⊢ ((dom ( I ↾ 𝐴) ⊆ 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ 𝐴) → (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴))
9	5, 7, 8	mp2an 708	. . 3 ⊢ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴)
10	3, 9	sstri 3612	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
11		dfhe2 38068	. 2 ⊢ ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
12	10, 11	mpbir 221	1 ⊢ I hereditary 𝐴

Syntax hints:   ⊆ wss 3574   I cid 5023   × cxp 5112  dom cdm 5114  ran crn 5115   ↾ cres 5116  Rel wrel 5119   hereditary whe 38066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by:  sshepw  38083