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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
StepHypRef Expression
1 relres 5634 . . . 4 Rel ( I ↾ 𝐴)
2 relssdmrn 5873 . . . 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 5674 . . . . 5 dom ( I ↾ 𝐴) = 𝐴
54eqimssi 3853 . . . 4 dom ( I ↾ 𝐴) ⊆ 𝐴
6 rnresi 5694 . . . . 5 ran ( I ↾ 𝐴) = 𝐴
76eqimssi 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))
95, 7, 8mp2an 684 . . 3 (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) ⊆ (𝐴 × 𝐴)
103, 9sstri 3805 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
11 dfhe2 38837 . 2 ( I hereditary 𝐴 ↔ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
1210, 11mpbir 223 1 I hereditary 𝐴
Colors of variables: wff setvar class
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
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