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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimrel | Structured version Visualization version GIF version |
Description: The limit on extended reals is a relation. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimrel | ⊢ Rel ~~>* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmrel 21766 | . 2 ⊢ Rel (⇝𝑡‘(ordTop‘ ≤ )) | |
2 | df-xlim 41976 | . . 3 ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | |
3 | 2 | releqi 5645 | . 2 ⊢ (Rel ~~>* ↔ Rel (⇝𝑡‘(ordTop‘ ≤ ))) |
4 | 1, 3 | mpbir 232 | 1 ⊢ Rel ~~>* |
Colors of variables: wff setvar class |
Syntax hints: Rel wrel 5553 ‘cfv 6348 ≤ cle 10664 ordTopcordt 16760 ⇝𝑡clm 21762 ~~>*clsxlim 41975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-lm 21765 df-xlim 41976 |
This theorem is referenced by: dmclimxlim 42008 xlimresdm 42016 |
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