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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 32323 with a different symbol, cf. https://github.com/metamath/set.mm/issues/2469. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
xrnrel | ⊢ Rel (𝐴 ⋉ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnss3v 34474 | . . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V)) | |
2 | xpss 5266 | . . 3 ⊢ (V × (V × V)) ⊆ (V × V) | |
3 | 1, 2 | sstri 3761 | . 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V) |
4 | df-rel 5257 | . 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V)) | |
5 | 3, 4 | mpbir 221 | 1 ⊢ Rel (𝐴 ⋉ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3351 ⊆ wss 3723 × cxp 5248 Rel wrel 5255 ⋉ cxrn 34312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-br 4788 df-opab 4848 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-res 5262 df-xrn 34473 |
This theorem is referenced by: dfxrn2 34478 elecxrn 34488 inxpxrn 34493 br1cnvxrn2 34494 |
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