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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 6008 | . . 3 ⊢ (𝑆 ⊆ 𝑅 → (𝑆 “ 𝐴) ⊆ (𝑅 “ 𝐴)) | |
2 | sstr2 3933 | . . 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 41363 | . 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴) | |
5 | df-he 41363 | . 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 3892 “ cima 5593 hereditary whe 41362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-cnv 5598 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-he 41363 |
This theorem is referenced by: (None) |
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