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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrnrel | Structured version Visualization version GIF version |
Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 32951 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 32951. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
xrnrel | ⊢ Rel (𝐴 ⋉ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnss3v 35176 | . . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V)) | |
2 | xpss 5466 | . . 3 ⊢ (V × (V × V)) ⊆ (V × V) | |
3 | 1, 2 | sstri 3904 | . 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V) |
4 | df-rel 5457 | . 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V)) | |
5 | 3, 4 | mpbir 232 | 1 ⊢ Rel (𝐴 ⋉ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3440 ⊆ wss 3865 × cxp 5448 Rel wrel 5455 ⋉ cxrn 35005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-br 4969 df-opab 5031 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-res 5462 df-xrn 35175 |
This theorem is referenced by: dfxrn2 35180 elecxrn 35190 inxpxrn 35195 br1cnvxrn2 35196 disjxrn 35529 |
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