Theorem hess 44397

Description: Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)

Assertion

Ref	Expression
hess	⊢ (𝑆 ⊆ 𝑅 → (𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴))

Proof of Theorem hess

Step	Hyp	Ref	Expression
1		imass1 6104	. . 3 ⊢ (𝑆 ⊆ 𝑅 → (𝑆 “ 𝐴) ⊆ (𝑅 “ 𝐴))
2		sstr2 3952	. . 3 ⊢ ((𝑆 “ 𝐴) ⊆ (𝑅 “ 𝐴) → ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝑆 “ 𝐴) ⊆ 𝐴))
3	1, 2	syl 18	. 2 ⊢ (𝑆 ⊆ 𝑅 → ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝑆 “ 𝐴) ⊆ 𝐴))
4		df-he 44390	. 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
5		df-he 44390	. 2 ⊢ (𝑆 hereditary 𝐴 ↔ (𝑆 “ 𝐴) ⊆ 𝐴)
6	3, 4, 5	3imtr4g 299	1 ⊢ (𝑆 ⊆ 𝑅 → (𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3913   “ cima 5665   hereditary whe 44389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-he 44390

This theorem is referenced by: (None)