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Theorem hess 43769
Description: Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
Assertion
Ref Expression
hess (𝑆𝑅 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))

Proof of Theorem hess
StepHypRef Expression
1 imass1 6121 . . 3 (𝑆𝑅 → (𝑆𝐴) ⊆ (𝑅𝐴))
2 sstr2 4001 . . 3 ((𝑆𝐴) ⊆ (𝑅𝐴) → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
31, 2syl 17 . 2 (𝑆𝑅 → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
4 df-he 43762 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
5 df-he 43762 . 2 (𝑆 hereditary 𝐴 ↔ (𝑆𝐴) ⊆ 𝐴)
63, 4, 53imtr4g 296 1 (𝑆𝑅 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3962  cima 5691   hereditary whe 43761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-he 43762
This theorem is referenced by: (None)
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