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Mirrors > Home > MPE Home > Th. List > metf | Structured version Visualization version Unicode version |
Description: Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) |
Ref | Expression |
---|---|
metf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metflem 22133 | . 2 | |
2 | 1 | simpld 475 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 class class class wbr 4653 cxp 5112 wf 5884 cfv 5888 (class class class)co 6650 cr 9935 cc0 9936 caddc 9939 cle 10075 cme 19732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-met 19740 |
This theorem is referenced by: metcl 22137 metn0 22165 metres2 22168 metres 22170 msf 22263 isngp3 22402 tngngp2 22456 tngngpim 22463 xrsdsre 22613 metdcn2 22642 cncms 23151 cnrrext 30054 isbnd3 33583 isbnd3b 33584 ssbnd 33587 bnd2lem 33590 prdsbnd 33592 |
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