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Mirrors > Home > MPE Home > Th. List > trrelssd | Structured version Visualization version GIF version |
Description: The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
trrelssd.r | ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
trrelssd.s | ⊢ (𝜑 → 𝑆 ⊆ 𝑅) |
trrelssd.t | ⊢ (𝜑 → 𝑇 ⊆ 𝑅) |
Ref | Expression |
---|---|
trrelssd | ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trrelssd.s | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑅) | |
2 | trrelssd.t | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑅) | |
3 | 1, 2 | coss12d 14943 | . 2 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ (𝑅 ∘ 𝑅)) |
4 | trrelssd.r | . 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
5 | 3, 4 | sstrd 3988 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3944 ∘ ccom 5676 |
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-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3471 df-in 3951 df-ss 3961 df-br 5143 df-opab 5205 df-co 5681 |
This theorem is referenced by: trclfvlb2 14981 trrelind 43018 iunrelexpmin1 43061 iunrelexpmin2 43065 |
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