Theorem unhe1 44569

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 44557	. . 3 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
2		df-he 44557	. . 3 ⊢ (𝑆 hereditary 𝐴 ↔ (𝑆 “ 𝐴) ⊆ 𝐴)
3		imaundir 6150	. . . 4 ⊢ ((𝑅 ∪ 𝑆) “ 𝐴) = ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴))
4		unss 4143	. . . . 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 3976	. . 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 44557	. 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   ∪ cun 3904   ⊆ wss 3906   “ cima 5666   hereditary whe 44556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-he 44557

This theorem is used by:  sshepw  44573