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Mirrors > Home > MPE Home > Th. List > Mathboxes > trrelind | Structured version Visualization version GIF version |
Description: The intersection of transitive relations is a transitive relation. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
trrelind.r | ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
trrelind.s | ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
trrelind.t | ⊢ (𝜑 → 𝑇 = (𝑅 ∩ 𝑆)) |
Ref | Expression |
---|---|
trrelind | ⊢ (𝜑 → (𝑇 ∘ 𝑇) ⊆ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trrelind.r | . . . 4 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
2 | inss1 4155 | . . . . 5 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑅 | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑅 ∩ 𝑆) ⊆ 𝑅) |
4 | 1, 3, 3 | trrelssd 14324 | . . 3 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ 𝑅) |
5 | trrelind.s | . . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) | |
6 | inss2 4156 | . . . . 5 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑆 | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑅 ∩ 𝑆) ⊆ 𝑆) |
8 | 5, 7, 7 | trrelssd 14324 | . . 3 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ 𝑆) |
9 | 4, 8 | ssind 4159 | . 2 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆)) |
10 | trrelind.t | . . 3 ⊢ (𝜑 → 𝑇 = (𝑅 ∩ 𝑆)) | |
11 | 10, 10 | coeq12d 5699 | . 2 ⊢ (𝜑 → (𝑇 ∘ 𝑇) = ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆))) |
12 | 9, 11, 10 | 3sstr4d 3962 | 1 ⊢ (𝜑 → (𝑇 ∘ 𝑇) ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∩ cin 3880 ⊆ wss 3881 ∘ ccom 5523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-br 5031 df-opab 5093 df-co 5528 |
This theorem is referenced by: xpintrreld 40367 |
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