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Mirrors > Home > MPE Home > Th. List > Mathboxes > xpintrreld | Structured version Visualization version GIF version |
Description: The intersection of a transitive relation with a cross product is a transitve relation. (Contributed by Richard Penner, 24-Dec-2019.) |
Ref | Expression |
---|---|
xpintrreld.r | ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
xpintrreld.s | ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × 𝐵))) |
Ref | Expression |
---|---|
xpintrreld | ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpintrreld.r | . 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
2 | xptrrel 14201 | . . 3 ⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)) |
4 | xpintrreld.s | . 2 ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × 𝐵))) | |
5 | 1, 3, 4 | trrelind 39379 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∩ cin 3828 ⊆ wss 3829 × cxp 5405 ∘ ccom 5411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-br 4930 df-opab 4992 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 |
This theorem is referenced by: restrreld 39381 |
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