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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 5535 | . 2 ⊢ Rel (𝐴 × 𝐵) | |
2 | relin2 5648 | . 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 3853 × cxp 5515 Rel wrel 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-rab 3077 df-v 3409 df-in 3861 df-ss 3871 df-opab 5088 df-xp 5523 df-rel 5524 |
This theorem is referenced by: elinxp 5854 cnvcnv 6014 tpostpos 7915 erinxp 8374 brdom3 9973 brdom5 9974 brdom4 9975 fpwwe2lem7 10082 fpwwe2lem8 10083 fpwwe2lem11 10086 pwsle 16808 opsrtoslem2 20801 elrn3 33230 bj-idres 34840 inxprnres 35974 inxpss 35994 inxpss2 35997 iss2 36026 inxp2 36044 inxpxrn 36068 |
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