Theorem unhe1 43781

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 43769	. . 3 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
2		df-he 43769	. . 3 ⊢ (𝑆 hereditary 𝐴 ↔ (𝑆 “ 𝐴) ⊆ 𝐴)
3		imaundir 6126	. . . 4 ⊢ ((𝑅 ∪ 𝑆) “ 𝐴) = ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴))
4		unss 4156	. . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) ↔ ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴)) ⊆ 𝐴)
5	4	biimpi 216	. . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) → ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴)) ⊆ 𝐴)
6	3, 5	eqsstrid 3988	. . 3 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) → ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴)
7	1, 2, 6	syl2anb 598	. 2 ⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴)
8		df-he 43769	. 2 ⊢ ((𝑅 ∪ 𝑆) hereditary 𝐴 ↔ ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴)
9	7, 8	sylibr 234	1 ⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → (𝑅 ∪ 𝑆) hereditary 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∪ cun 3915   ⊆ wss 3917   “ cima 5644   hereditary whe 43768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-he 43769

This theorem is referenced by:  sshepw  43785