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Theorem hess 39030
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 5754 . . 3 (𝑆𝑅 → (𝑆𝐴) ⊆ (𝑅𝐴))
2 sstr2 3828 . . 3 ((𝑆𝐴) ⊆ (𝑅𝐴) → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
31, 2syl 17 . 2 (𝑆𝑅 → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
4 df-he 39023 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
5 df-he 39023 . 2 (𝑆 hereditary 𝐴 ↔ (𝑆𝐴) ⊆ 𝐴)
63, 4, 53imtr4g 288 1 (𝑆𝑅 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3792  cima 5358   hereditary whe 39022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-he 39023
This theorem is referenced by: (None)
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