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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 3222 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
2 | relss 4538 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 → (Rel 𝐵 → Rel (𝐴 ∩ 𝐵))) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (Rel 𝐵 → Rel (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∩ cin 2999 ⊆ wss 3000 Rel wrel 4457 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-in 3006 df-ss 3013 df-rel 4459 |
This theorem is referenced by: intasym 4829 asymref 4830 poirr2 4837 |
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