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Mirrors > Home > MPE Home > Th. List > relinxp | Structured version Visualization version GIF version |
Description: Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.) |
Ref | Expression |
---|---|
relinxp | ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5330 | . 2 ⊢ Rel (𝐴 × 𝐵) | |
2 | relin2 5440 | . 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3768 × cxp 5310 Rel wrel 5317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-v 3387 df-in 3776 df-ss 3783 df-opab 4906 df-xp 5318 df-rel 5319 |
This theorem is referenced by: elinxp 5644 erinxp 8059 fpwwe2lem8 9747 fpwwe2lem9 9748 fpwwe2lem12 9751 pwsle 16467 inxpss 34577 inxpss2 34580 iss2 34606 inxp2 34623 inxpxrn 34647 |
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