Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > relin2 | 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 3267 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
2 | relss 4596 | . 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 set class |
Syntax hints: → wi 4 ∩ cin 3040 ⊆ wss 3041 Rel wrel 4514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 df-rel 4516 |
This theorem is referenced by: intasym 4893 asymref 4894 poirr2 4901 |
Copyright terms: Public domain | W3C validator |