Theorem idhe 38850

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 5634	. . . 4 ⊢ Rel ( I ↾ 𝐴)
2		relssdmrn 5873	. . . 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 5674	. . . . 5 ⊢ dom ( I ↾ 𝐴) = 𝐴
5	4	eqimssi 3853	. . . 4 ⊢ dom ( I ↾ 𝐴) ⊆ 𝐴
6		rnresi 5694	. . . . 5 ⊢ ran ( I ↾ 𝐴) = 𝐴
7	6	eqimssi 3853	. . . 4 ⊢ ran ( I ↾ 𝐴) ⊆ 𝐴
8		xpss12 5325	. . . 4 ⊢ ((dom ( I ↾ 𝐴) ⊆ 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ 𝐴) → (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴))
9	5, 7, 8	mp2an 684	. . 3 ⊢ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴)
10	3, 9	sstri 3805	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
11		dfhe2 38837	. 2 ⊢ ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
12	10, 11	mpbir 223	1 ⊢ I hereditary 𝐴

Syntax hints:   ⊆ wss 3767   I cid 5217   × cxp 5308  dom cdm 5310  ran crn 5311   ↾ cres 5312  Rel wrel 5315   hereditary whe 38835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-he 38836

This theorem is referenced by:  sshepw  38852