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Theorem dfhe2 37589
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 37588 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 rp-imass 37586 . 2 ((𝑅𝐴) ⊆ 𝐴 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐴))
31, 2bitri 264 1 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wss 3560   × cxp 5082  cres 5086  cima 5087   hereditary whe 37587
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 4751  ax-nul 4759  ax-pr 4877
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-he 37588
This theorem is referenced by:  idhe  37602
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