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Theorem unhe1 41374
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 41362 . . 3 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 df-he 41362 . . 3 (𝑆 hereditary 𝐴 ↔ (𝑆𝐴) ⊆ 𝐴)
3 imaundir 6047 . . . 4 ((𝑅𝑆) “ 𝐴) = ((𝑅𝐴) ∪ (𝑆𝐴))
4 unss 4117 . . . . 5 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) ↔ ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
54biimpi 215 . . . 4 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
63, 5eqsstrid 3968 . . 3 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
71, 2, 6syl2anb 598 . 2 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
8 df-he 41362 . 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 396  cun 3884  wss 3886  cima 5587   hereditary whe 41361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5074  df-opab 5136  df-cnv 5592  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-he 41362
This theorem is referenced by:  sshepw  41378
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