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Theorem hess 40568
 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 5934 . . 3 (𝑆𝑅 → (𝑆𝐴) ⊆ (𝑅𝐴))
2 sstr2 3923 . . 3 ((𝑆𝐴) ⊆ (𝑅𝐴) → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
31, 2syl 17 . 2 (𝑆𝑅 → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
4 df-he 40561 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
5 df-he 40561 . 2 (𝑆 hereditary 𝐴 ↔ (𝑆𝐴) ⊆ 𝐴)
63, 4, 53imtr4g 299 1 (𝑆𝑅 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3882   “ cima 5525   hereditary whe 40560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3887  df-in 3889  df-ss 3899  df-sn 4528  df-pr 4530  df-op 4534  df-br 5034  df-opab 5096  df-cnv 5530  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-he 40561 This theorem is referenced by: (None)
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