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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 15031 | . . 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 43629 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3975 ⊆ wss 3976 × cxp 5698 ∘ ccom 5704 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
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 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 |
This theorem is referenced by: restrreld 43631 |
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