Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 14681 | . 2 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ (𝑅 ∘ 𝑅)) |
4 | trrelssd.r | . 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
5 | 3, 4 | sstrd 3932 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3888 ∘ ccom 5595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3433 df-in 3895 df-ss 3905 df-br 5077 df-opab 5139 df-co 5600 |
This theorem is referenced by: trclfvlb2 14719 trrelind 41243 iunrelexpmin1 41286 iunrelexpmin2 41290 |
Copyright terms: Public domain | W3C validator |