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Mirrors > Home > MPE Home > Th. List > Mathboxes > unhe1 | Structured version Visualization version GIF version |
Description: The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.) |
Ref | Expression |
---|---|
unhe1 | ⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → (𝑅 ∪ 𝑆) hereditary 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-he 43203 | . . 3 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴) | |
2 | df-he 43203 | . . 3 ⊢ (𝑆 hereditary 𝐴 ↔ (𝑆 “ 𝐴) ⊆ 𝐴) | |
3 | imaundir 6155 | . . . 4 ⊢ ((𝑅 ∪ 𝑆) “ 𝐴) = ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴)) | |
4 | unss 4184 | . . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) ↔ ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴)) ⊆ 𝐴) | |
5 | 4 | biimpi 215 | . . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) → ((𝑅 “ 𝐴) ∪ (𝑆 “ 𝐴)) ⊆ 𝐴) |
6 | 3, 5 | eqsstrid 4028 | . . 3 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ (𝑆 “ 𝐴) ⊆ 𝐴) → ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴) |
7 | 1, 2, 6 | syl2anb 597 | . 2 ⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴) |
8 | df-he 43203 | . 2 ⊢ ((𝑅 ∪ 𝑆) hereditary 𝐴 ↔ ((𝑅 ∪ 𝑆) “ 𝐴) ⊆ 𝐴) | |
9 | 7, 8 | sylibr 233 | 1 ⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → (𝑅 ∪ 𝑆) hereditary 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∪ cun 3945 ⊆ wss 3947 “ cima 5681 hereditary whe 43202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-he 43203 |
This theorem is referenced by: sshepw 43219 |
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