Theorem idhe 37101

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 5346	. . . 4 ⊢ Rel ( I ↾ 𝐴)
2		relssdmrn 5573	. . . 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 5376	. . . . 5 ⊢ dom ( I ↾ 𝐴) = 𝐴
5	4	eqimssi 3622	. . . 4 ⊢ dom ( I ↾ 𝐴) ⊆ 𝐴
6		rnresi 5398	. . . . 5 ⊢ ran ( I ↾ 𝐴) = 𝐴
7	6	eqimssi 3622	. . . 4 ⊢ ran ( I ↾ 𝐴) ⊆ 𝐴
8		xpss12 5148	. . . 4 ⊢ ((dom ( I ↾ 𝐴) ⊆ 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ 𝐴) → (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴))
9	5, 7, 8	mp2an 704	. . 3 ⊢ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴)
10	3, 9	sstri 3577	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
11		dfhe2 37088	. 2 ⊢ ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
12	10, 11	mpbir 220	1 ⊢ I hereditary 𝐴

Syntax hints:   ⊆ wss 3540   I cid 4948   × cxp 5036  dom cdm 5038  ran crn 5039   ↾ cres 5040  Rel wrel 5043   hereditary whe 37086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-he 37087

This theorem is referenced by:  sshepw  37103