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Mirrors > Home > ILE Home > Th. List > xmetunirn | Unicode version |
Description: Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
xmetunirn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmap 6549 | . . . . . . 7 | |
2 | xrex 9639 | . . . . . . 7 | |
3 | sqxpexg 4655 | . . . . . . . 8 | |
4 | 3 | elv 2690 | . . . . . . 7 |
5 | fnovex 5804 | . . . . . . 7 | |
6 | 1, 2, 4, 5 | mp3an 1315 | . . . . . 6 |
7 | 6 | rabex 4072 | . . . . 5 |
8 | df-xmet 12157 | . . . . 5 | |
9 | 7, 8 | fnmpti 5251 | . . . 4 |
10 | fnunirn 5668 | . . . 4 | |
11 | 9, 10 | ax-mp 5 | . . 3 |
12 | id 19 | . . . . 5 | |
13 | xmetdmdm 12525 | . . . . . 6 | |
14 | 13 | fveq2d 5425 | . . . . 5 |
15 | 12, 14 | eleqtrd 2218 | . . . 4 |
16 | 15 | rexlimivw 2545 | . . 3 |
17 | 11, 16 | sylbi 120 | . 2 |
18 | elex 2697 | . . . . . 6 | |
19 | dmexg 4803 | . . . . . 6 | |
20 | dmexg 4803 | . . . . . 6 | |
21 | 18, 19, 20 | 3syl 17 | . . . . 5 |
22 | fvssunirng 5436 | . . . . 5 | |
23 | 21, 22 | syl 14 | . . . 4 |
24 | 23 | sseld 3096 | . . 3 |
25 | 24 | pm2.43i 49 | . 2 |
26 | 17, 25 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 crab 2420 cvv 2686 wss 3071 cuni 3736 class class class wbr 3929 cxp 4537 cdm 4539 crn 4540 wfn 5118 cfv 5123 (class class class)co 5774 cmap 6542 cc0 7620 cxr 7799 cle 7801 cxad 9557 cxmet 12149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-xmet 12157 |
This theorem is referenced by: isxms2 12621 |
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