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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 5687 | . . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
4 | 2, 3 | eqtrdi 2789 | . 2 ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × V))) |
5 | 1, 4 | xpintrreld 42350 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Vcvv 3475 ∩ cin 3946 ⊆ wss 3947 × cxp 5673 ↾ cres 5677 ∘ ccom 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 |
This theorem is referenced by: (None) |
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