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Theorem hess 37591
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 5464 . . 3 (𝑆𝑅 → (𝑆𝐴) ⊆ (𝑅𝐴))
2 sstr2 3594 . . 3 ((𝑆𝐴) ⊆ (𝑅𝐴) → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
31, 2syl 17 . 2 (𝑆𝑅 → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
4 df-he 37584 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
5 df-he 37584 . 2 (𝑆 hereditary 𝐴 ↔ (𝑆𝐴) ⊆ 𝐴)
63, 4, 53imtr4g 285 1 (𝑆𝑅 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3559  cima 5082   hereditary whe 37583
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
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-he 37584
This theorem is referenced by: (None)
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