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Theorem unhe1 43486
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 43474 . . 3 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 df-he 43474 . . 3 (𝑆 hereditary 𝐴 ↔ (𝑆𝐴) ⊆ 𝐴)
3 imaundir 6152 . . . 4 ((𝑅𝑆) “ 𝐴) = ((𝑅𝐴) ∪ (𝑆𝐴))
4 unss 4182 . . . . 5 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) ↔ ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
54biimpi 215 . . . 4 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
63, 5eqsstrid 4027 . . 3 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
71, 2, 6syl2anb 596 . 2 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
8 df-he 43474 . 2 ((𝑅𝑆) hereditary 𝐴 ↔ ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
97, 8sylibr 233 1 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → (𝑅𝑆) hereditary 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  cun 3944  wss 3946  cima 5675   hereditary whe 43473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5144  df-opab 5206  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-he 43474
This theorem is referenced by:  sshepw  43490
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