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Theorem dfhe2 43074
Description: The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
Assertion
Ref Expression
dfhe2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐴))

Proof of Theorem dfhe2
StepHypRef Expression
1 df-he 43073 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 resssxp 6260 . 2 ((𝑅𝐴) ⊆ 𝐴 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐴))
31, 2bitri 275 1 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wss 3941   × cxp 5665  cres 5669  cima 5670   hereditary whe 43072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-he 43073
This theorem is referenced by:  idhe  43087
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