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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 4204 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | relss 5650 | . 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 3934 ⊆ wss 3935 Rel wrel 5554 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3942 df-ss 3951 df-rel 5556 |
This theorem is referenced by: inopab 5695 idsset 33346 dihmeetlem1N 38420 dihglblem5apreN 38421 dihmeetlem4preN 38436 dihmeetlem13N 38449 |
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