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Mirrors > Home > ILE Home > Th. List > ismet | Unicode version |
Description: Express the predicate " is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
ismet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2692 | . . . . 5 | |
2 | fnmap 6542 | . . . . . . . 8 | |
3 | reex 7747 | . . . . . . . 8 | |
4 | sqxpexg 4650 | . . . . . . . 8 | |
5 | fnovex 5797 | . . . . . . . 8 | |
6 | 2, 3, 4, 5 | mp3an12i 1319 | . . . . . . 7 |
7 | rabexg 4066 | . . . . . . 7 | |
8 | 6, 7 | syl 14 | . . . . . 6 |
9 | xpeq12 4553 | . . . . . . . . . 10 | |
10 | 9 | anidms 394 | . . . . . . . . 9 |
11 | 10 | oveq2d 5783 | . . . . . . . 8 |
12 | raleq 2624 | . . . . . . . . . . 11 | |
13 | 12 | anbi2d 459 | . . . . . . . . . 10 |
14 | 13 | raleqbi1dv 2632 | . . . . . . . . 9 |
15 | 14 | raleqbi1dv 2632 | . . . . . . . 8 |
16 | 11, 15 | rabeqbidv 2676 | . . . . . . 7 |
17 | df-met 12147 | . . . . . . 7 | |
18 | 16, 17 | fvmptg 5490 | . . . . . 6 |
19 | 8, 18 | mpdan 417 | . . . . 5 |
20 | 1, 19 | syl 14 | . . . 4 |
21 | 20 | eleq2d 2207 | . . 3 |
22 | oveq 5773 | . . . . . . . 8 | |
23 | 22 | eqeq1d 2146 | . . . . . . 7 |
24 | 23 | bibi1d 232 | . . . . . 6 |
25 | oveq 5773 | . . . . . . . . 9 | |
26 | oveq 5773 | . . . . . . . . 9 | |
27 | 25, 26 | oveq12d 5785 | . . . . . . . 8 |
28 | 22, 27 | breq12d 3937 | . . . . . . 7 |
29 | 28 | ralbidv 2435 | . . . . . 6 |
30 | 24, 29 | anbi12d 464 | . . . . 5 |
31 | 30 | 2ralbidv 2457 | . . . 4 |
32 | 31 | elrab 2835 | . . 3 |
33 | 21, 32 | syl6bb 195 | . 2 |
34 | sqxpexg 4650 | . . . 4 | |
35 | elmapg 6548 | . . . 4 | |
36 | 3, 34, 35 | sylancr 410 | . . 3 |
37 | 36 | anbi1d 460 | . 2 |
38 | 33, 37 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 crab 2418 cvv 2681 class class class wbr 3924 cxp 4532 wfn 5113 wf 5114 cfv 5118 (class class class)co 5767 cmap 6535 cr 7612 cc0 7613 caddc 7616 cle 7794 cmet 12139 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-map 6537 df-met 12147 |
This theorem is referenced by: ismeti 12504 metflem 12507 ismet2 12512 |
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