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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 Richard Penner, 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 3795 | . . . . 5 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑅 | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑅 ∩ 𝑆) ⊆ 𝑅) |
4 | 1, 3, 3 | trrelssd 13560 | . . 3 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ 𝑅) |
5 | trrelind.s | . . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) | |
6 | inss2 3796 | . . . . 5 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑆 | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑅 ∩ 𝑆) ⊆ 𝑆) |
8 | 5, 7, 7 | trrelssd 13560 | . . 3 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ 𝑆) |
9 | 4, 8 | ssind 3799 | . 2 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆)) |
10 | trrelind.t | . . 3 ⊢ (𝜑 → 𝑇 = (𝑅 ∩ 𝑆)) | |
11 | 10, 10 | coeq12d 5208 | . 2 ⊢ (𝜑 → (𝑇 ∘ 𝑇) = ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆))) |
12 | 9, 11, 10 | 3sstr4d 3611 | 1 ⊢ (𝜑 → (𝑇 ∘ 𝑇) ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∩ cin 3539 ⊆ wss 3540 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-br 4584 df-opab 4644 df-co 5047 |
This theorem is referenced by: xpintrreld 36977 |
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