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Mirrors > Home > MPE Home > Th. List > Mathboxes > hess | Structured version Visualization version GIF version |
Description: Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
Ref | Expression |
---|---|
hess | ⊢ (𝑆 ⊆ 𝑅 → (𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imass1 6121 | . . 3 ⊢ (𝑆 ⊆ 𝑅 → (𝑆 “ 𝐴) ⊆ (𝑅 “ 𝐴)) | |
2 | sstr2 4001 | . . 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 43762 | . 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴) | |
5 | df-he 43762 | . 2 ⊢ (𝑆 hereditary 𝐴 ↔ (𝑆 “ 𝐴) ⊆ 𝐴) | |
6 | 3, 4, 5 | 3imtr4g 296 | 1 ⊢ (𝑆 ⊆ 𝑅 → (𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3962 “ cima 5691 hereditary whe 43761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-he 43762 |
This theorem is referenced by: (None) |
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