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Mirrors > Home > MPE Home > Th. List > relin1 | Structured version Visualization version GIF version |
Description: The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.) |
Ref | Expression |
---|---|
relin1 | ⊢ (Rel 𝐴 → Rel (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4155 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | relss 5620 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴 ∩ 𝐵))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Rel 𝐴 → Rel (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3880 ⊆ wss 3881 Rel wrel 5524 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-rel 5526 |
This theorem is referenced by: inopab 5665 idsset 33464 dihmeetlem1N 38586 dihglblem5apreN 38587 dihmeetlem4preN 38602 dihmeetlem13N 38615 |
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