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Mirrors > Home > MPE Home > Th. List > Mathboxes > restrreld | Structured version Visualization version GIF version |
Description: The restriction of a transitive relation is a transitive relation. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
restrreld.r | ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
restrreld.s | ⊢ (𝜑 → 𝑆 = (𝑅 ↾ 𝐴)) |
Ref | Expression |
---|---|
restrreld | ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restrreld.r | . 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
2 | restrreld.s | . . 3 ⊢ (𝜑 → 𝑆 = (𝑅 ↾ 𝐴)) | |
3 | df-res 5552 | . . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
4 | 2, 3 | eqtrdi 2790 | . 2 ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × V))) |
5 | 1, 4 | xpintrreld 40903 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 Vcvv 3401 ∩ cin 3856 ⊆ wss 3857 × cxp 5538 ↾ cres 5542 ∘ ccom 5544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 |
This theorem is referenced by: (None) |
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