Theorem unhe1 44894

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 𝐴 ∧ 𝑆 hereditary 𝐴) → (𝑅 ∪ 𝑆) hereditary 𝐴)

Proof of Theorem unhe1

Step	Hyp	Ref	Expression
1		df-he 44882	. . 3 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
2		df-he 44882	. . 3 ⊢ (𝑆 hereditary 𝐴 ↔ (𝑆 “ 𝐴) ⊆ 𝐴)
3		imaundir 6144	. . . 4 ⊢ ((𝑅 ∪ 𝑆) “ 𝐴) = ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴))
4		unss 4187	. . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) ↔ ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴)) ⊆ 𝐴)
5	4	biimpi 219	. . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) → ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴)) ⊆ 𝐴)
6	3, 5	eqsstrid 3947	. . 3 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) → ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴)
7	1, 2, 6	syl2anb 610	. 2 ⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴)
8		df-he 44882	. 2 ⊢ ((𝑅 ∪ 𝑆) hereditary 𝐴 ↔ ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴)
9	7, 8	sylibr 237	1 ⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → (𝑅 ∪ 𝑆) hereditary 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ⊆ wss 3896   ∪ cun 4148   “ cima 5654   hereditary whe 44881

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-he 44882

This theorem is used by:  sshepw  44898