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Theorem idhe 38398
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
StepHypRef Expression
1 relres 5461 . . . 4 Rel ( I ↾ 𝐴)
2 relssdmrn 5694 . . . 4 (Rel ( I ↾ 𝐴) → ( I ↾ 𝐴) ⊆ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)))
31, 2ax-mp 5 . . 3 ( I ↾ 𝐴) ⊆ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴))
4 dmresi 5492 . . . . 5 dom ( I ↾ 𝐴) = 𝐴
54eqimssi 3692 . . . 4 dom ( I ↾ 𝐴) ⊆ 𝐴
6 rnresi 5514 . . . . 5 ran ( I ↾ 𝐴) = 𝐴
76eqimssi 3692 . . . 4 ran ( I ↾ 𝐴) ⊆ 𝐴
8 xpss12 5158 . . . 4 ((dom ( I ↾ 𝐴) ⊆ 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ 𝐴) → (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴))
95, 7, 8mp2an 708 . . 3 (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴)
103, 9sstri 3645 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
11 dfhe2 38385 . 2 ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
1210, 11mpbir 221 1 I hereditary 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3607   I cid 5052   × cxp 5141  dom cdm 5143  ran crn 5144  cres 5145  Rel wrel 5148   hereditary whe 38383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-he 38384
This theorem is referenced by:  sshepw  38400
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