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Theorem unhe1 38849
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 38837 . . 3 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 df-he 38837 . . 3 (𝑆 hereditary 𝐴 ↔ (𝑆𝐴) ⊆ 𝐴)
3 imaundir 5761 . . . 4 ((𝑅𝑆) “ 𝐴) = ((𝑅𝐴) ∪ (𝑆𝐴))
4 unss 3983 . . . . 5 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) ↔ ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
54biimpi 208 . . . 4 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
63, 5syl5eqss 3843 . . 3 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
71, 2, 6syl2anb 592 . 2 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
8 df-he 38837 . 2 ((𝑅𝑆) hereditary 𝐴 ↔ ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
97, 8sylibr 226 1 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → (𝑅𝑆) hereditary 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  cun 3765  wss 3767  cima 5313   hereditary whe 38836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-cnv 5318  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-he 38837
This theorem is referenced by:  sshepw  38853
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