![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4229 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | relss 5783 | . 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 3946 ⊆ wss 3947 Rel wrel 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-in 3954 df-ss 3964 df-rel 5685 |
This theorem is referenced by: inopab 5831 idsset 35486 dihmeetlem1N 40763 dihglblem5apreN 40764 dihmeetlem4preN 40779 dihmeetlem13N 40792 |
Copyright terms: Public domain | W3C validator |