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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 Cartesian product is a transitive relation. (Contributed by RP, 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 15025 | . . 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 43568 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3969 ⊆ wss 3970 × cxp 5697 ∘ ccom 5703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 |
This theorem is referenced by: restrreld 43570 |
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