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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 4059 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | relss 5445 | . 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 3797 ⊆ wss 3798 Rel wrel 5351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 df-in 3805 df-ss 3812 df-rel 5353 |
This theorem is referenced by: inopab 5489 idsset 32531 dihmeetlem1N 37360 dihglblem5apreN 37361 dihmeetlem4preN 37376 dihmeetlem13N 37389 |
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