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Theorem hess 41370
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 6008 . . 3 (𝑆𝑅 → (𝑆𝐴) ⊆ (𝑅𝐴))
2 sstr2 3933 . . 3 ((𝑆𝐴) ⊆ (𝑅𝐴) → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
31, 2syl 17 . 2 (𝑆𝑅 → ((𝑅𝐴) ⊆ 𝐴 → (𝑆𝐴) ⊆ 𝐴))
4 df-he 41363 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
5 df-he 41363 . 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 3892  cima 5593   hereditary whe 41362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-cnv 5598  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-he 41363
This theorem is referenced by: (None)
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