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Theorem unhe1 40431
Description: The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)
Assertion
Ref Expression
unhe1 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → (𝑅𝑆) hereditary 𝐴)

Proof of Theorem unhe1
StepHypRef Expression
1 df-he 40419 . . 3 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 df-he 40419 . . 3 (𝑆 hereditary 𝐴 ↔ (𝑆𝐴) ⊆ 𝐴)
3 imaundir 5998 . . . 4 ((𝑅𝑆) “ 𝐴) = ((𝑅𝐴) ∪ (𝑆𝐴))
4 unss 4146 . . . . 5 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) ↔ ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
54biimpi 219 . . . 4 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
63, 5eqsstrid 4001 . . 3 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
71, 2, 6syl2anb 600 . 2 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
8 df-he 40419 . 2 ((𝑅𝑆) hereditary 𝐴 ↔ ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
97, 8sylibr 237 1 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → (𝑅𝑆) hereditary 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  cun 3917  wss 3919  cima 5546   hereditary whe 40418
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-cnv 5551  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-he 40419
This theorem is referenced by:  sshepw  40435
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