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Theorem idhe 37563
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 5385 . . . 4 Rel ( I ↾ 𝐴)
2 relssdmrn 5615 . . . 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 5416 . . . . 5 dom ( I ↾ 𝐴) = 𝐴
54eqimssi 3638 . . . 4 dom ( I ↾ 𝐴) ⊆ 𝐴
6 rnresi 5438 . . . . 5 ran ( I ↾ 𝐴) = 𝐴
76eqimssi 3638 . . . 4 ran ( I ↾ 𝐴) ⊆ 𝐴
8 xpss12 5186 . . . 4 ((dom ( I ↾ 𝐴) ⊆ 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ 𝐴) → (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴))
95, 7, 8mp2an 707 . . 3 (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴)
103, 9sstri 3592 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
11 dfhe2 37550 . 2 ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
1210, 11mpbir 221 1 I hereditary 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3555   I cid 4984   × cxp 5072  dom cdm 5074  ran crn 5075  cres 5076  Rel wrel 5079   hereditary whe 37548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-he 37549
This theorem is referenced by:  sshepw  37565
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