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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 4246 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
2 | relss 5794 | . 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 3962 ⊆ wss 3963 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-rel 5696 |
This theorem is referenced by: relinxp 5827 intasym 6138 asymref 6139 poirr2 6147 symgcom2 33087 cnvref4 38332 dfantisymrel4 38743 dfantisymrel5 38744 clcnvlem 43613 |
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