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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 2741 | . . . . 5 | |
2 | fnmap 6631 | . . . . . . . 8 | |
3 | reex 7901 | . . . . . . . 8 | |
4 | sqxpexg 4725 | . . . . . . . 8 | |
5 | fnovex 5884 | . . . . . . . 8 | |
6 | 2, 3, 4, 5 | mp3an12i 1336 | . . . . . . 7 |
7 | rabexg 4130 | . . . . . . 7 | |
8 | 6, 7 | syl 14 | . . . . . 6 |
9 | xpeq12 4628 | . . . . . . . . . 10 | |
10 | 9 | anidms 395 | . . . . . . . . 9 |
11 | 10 | oveq2d 5867 | . . . . . . . 8 |
12 | raleq 2665 | . . . . . . . . . . 11 | |
13 | 12 | anbi2d 461 | . . . . . . . . . 10 |
14 | 13 | raleqbi1dv 2673 | . . . . . . . . 9 |
15 | 14 | raleqbi1dv 2673 | . . . . . . . 8 |
16 | 11, 15 | rabeqbidv 2725 | . . . . . . 7 |
17 | df-met 12748 | . . . . . . 7 | |
18 | 16, 17 | fvmptg 5570 | . . . . . 6 |
19 | 8, 18 | mpdan 419 | . . . . 5 |
20 | 1, 19 | syl 14 | . . . 4 |
21 | 20 | eleq2d 2240 | . . 3 |
22 | oveq 5857 | . . . . . . . 8 | |
23 | 22 | eqeq1d 2179 | . . . . . . 7 |
24 | 23 | bibi1d 232 | . . . . . 6 |
25 | oveq 5857 | . . . . . . . . 9 | |
26 | oveq 5857 | . . . . . . . . 9 | |
27 | 25, 26 | oveq12d 5869 | . . . . . . . 8 |
28 | 22, 27 | breq12d 4000 | . . . . . . 7 |
29 | 28 | ralbidv 2470 | . . . . . 6 |
30 | 24, 29 | anbi12d 470 | . . . . 5 |
31 | 30 | 2ralbidv 2494 | . . . 4 |
32 | 31 | elrab 2886 | . . 3 |
33 | 21, 32 | bitrdi 195 | . 2 |
34 | sqxpexg 4725 | . . . 4 | |
35 | elmapg 6637 | . . . 4 | |
36 | 3, 34, 35 | sylancr 412 | . . 3 |
37 | 36 | anbi1d 462 | . 2 |
38 | 33, 37 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 crab 2452 cvv 2730 class class class wbr 3987 cxp 4607 wfn 5191 wf 5192 cfv 5196 (class class class)co 5851 cmap 6624 cr 7766 cc0 7767 caddc 7770 cle 7948 cmet 12740 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-map 6626 df-met 12748 |
This theorem is referenced by: ismeti 13105 metflem 13108 ismet2 13113 |
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