Theorem hess 43773

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 6056	. . 3 ⊢ (𝑆 ⊆ 𝑅 → (𝑆 “ 𝐴) ⊆ (𝑅 “ 𝐴))
2		sstr2 3944	. . 3 ⊢ ((𝑆 “ 𝐴) ⊆ (𝑅 “ 𝐴) → ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝑆 “ 𝐴) ⊆ 𝐴))
3	1, 2	syl 17	. 2 ⊢ (𝑆 ⊆ 𝑅 → ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝑆 “ 𝐴) ⊆ 𝐴))
4		df-he 43766	. 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
5		df-he 43766	. 2 ⊢ (𝑆 hereditary 𝐴 ↔ (𝑆 “ 𝐴) ⊆ 𝐴)
6	3, 4, 5	3imtr4g 296	1 ⊢ (𝑆 ⊆ 𝑅 → (𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3905   “ cima 5626   hereditary whe 43765

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-ext 2701

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-he 43766

This theorem is referenced by: (None)