Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unhe1 Structured version   Visualization version   GIF version

Theorem unhe1 37558
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 37546 . . 3 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 df-he 37546 . . 3 (𝑆 hereditary 𝐴 ↔ (𝑆𝐴) ⊆ 𝐴)
3 imaundir 5505 . . . 4 ((𝑅𝑆) “ 𝐴) = ((𝑅𝐴) ∪ (𝑆𝐴))
4 unss 3765 . . . . 5 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) ↔ ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
54biimpi 206 . . . 4 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝐴) ∪ (𝑆𝐴)) ⊆ 𝐴)
63, 5syl5eqss 3628 . . 3 (((𝑅𝐴) ⊆ 𝐴 ∧ (𝑆𝐴) ⊆ 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
71, 2, 6syl2anb 496 . 2 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
8 df-he 37546 . 2 ((𝑅𝑆) hereditary 𝐴 ↔ ((𝑅𝑆) “ 𝐴) ⊆ 𝐴)
97, 8sylibr 224 1 ((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → (𝑅𝑆) hereditary 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  cun 3553  wss 3555  cima 5077   hereditary whe 37545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-he 37546
This theorem is referenced by:  sshepw  37562
  Copyright terms: Public domain W3C validator