Theorem hess 44917

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 3916	. . 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 44910	. 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
5		df-he 44910	. 2 ⊢ (𝑆 hereditary 𝐴 ↔ (𝑆 “ 𝐴) ⊆ 𝐴)
6	3, 4, 5	3imtr4g 299	1 ⊢ (𝑆 ⊆ 𝑅 → (𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴))

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3896   “ cima 5654   hereditary whe 44909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-he 44910

This theorem is used by: (None)