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| 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 4237 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
| 2 | relss 5791 | . 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 3950 ⊆ wss 3951 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-ss 3968 df-rel 5692 | 
| This theorem is referenced by: inopab 5839 idsset 35891 dihmeetlem1N 41292 dihglblem5apreN 41293 dihmeetlem4preN 41308 dihmeetlem13N 41321 thincciso2 49104 | 
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