Theorem unhe1 41393

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 41381	. . 3 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
2		df-he 41381	. . 3 ⊢ (𝑆 hereditary 𝐴 ↔ (𝑆 “ 𝐴) ⊆ 𝐴)
3		imaundir 6054	. . . 4 ⊢ ((𝑅 ∪ 𝑆) “ 𝐴) = ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴))
4		unss 4118	. . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) ↔ ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴)) ⊆ 𝐴)
5	4	biimpi 215	. . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) → ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴)) ⊆ 𝐴)
6	3, 5	eqsstrid 3969	. . 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 41381	. 2 ⊢ ((𝑅 ∪ 𝑆) hereditary 𝐴 ↔ ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴)
9	7, 8	sylibr 233	1 ⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → (𝑅 ∪ 𝑆) hereditary 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∪ cun 3885   ⊆ wss 3887   “ cima 5592   hereditary whe 41380

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 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-he 41381

This theorem is referenced by:  sshepw  41397