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Mirrors > Home > MPE Home > Th. List > relin2 | Structured version Visualization version GIF version |
Description: The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.) |
Ref | Expression |
---|---|
relin2 | ⊢ (Rel 𝐵 → Rel (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 4187 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
2 | relss 5735 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 → (Rel 𝐵 → Rel (𝐴 ∩ 𝐵))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Rel 𝐵 → Rel (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3907 ⊆ wss 3908 Rel wrel 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-in 3915 df-ss 3925 df-rel 5638 |
This theorem is referenced by: relinxp 5768 intasym 6067 asymref 6068 poirr2 6076 symgcom2 31777 cnvref4 36743 dfantisymrel4 37155 dfantisymrel5 37156 clcnvlem 41800 |
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