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Mirrors > Home > ILE Home > Th. List > isxmet | Unicode version |
Description: Express the predicate " is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
isxmet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2671 | . . . . 5 | |
2 | fnmap 6517 | . . . . . . . 8 | |
3 | xrex 9607 | . . . . . . . 8 | |
4 | sqxpexg 4625 | . . . . . . . 8 | |
5 | fnovex 5772 | . . . . . . . 8 | |
6 | 2, 3, 4, 5 | mp3an12i 1304 | . . . . . . 7 |
7 | rabexg 4041 | . . . . . . 7 | |
8 | 6, 7 | syl 14 | . . . . . 6 |
9 | xpeq12 4528 | . . . . . . . . . 10 | |
10 | 9 | anidms 394 | . . . . . . . . 9 |
11 | 10 | oveq2d 5758 | . . . . . . . 8 |
12 | raleq 2603 | . . . . . . . . . . 11 | |
13 | 12 | anbi2d 459 | . . . . . . . . . 10 |
14 | 13 | raleqbi1dv 2611 | . . . . . . . . 9 |
15 | 14 | raleqbi1dv 2611 | . . . . . . . 8 |
16 | 11, 15 | rabeqbidv 2655 | . . . . . . 7 |
17 | df-xmet 12084 | . . . . . . 7 | |
18 | 16, 17 | fvmptg 5465 | . . . . . 6 |
19 | 8, 18 | mpdan 417 | . . . . 5 |
20 | 1, 19 | syl 14 | . . . 4 |
21 | 20 | eleq2d 2187 | . . 3 |
22 | oveq 5748 | . . . . . . . 8 | |
23 | 22 | eqeq1d 2126 | . . . . . . 7 |
24 | 23 | bibi1d 232 | . . . . . 6 |
25 | oveq 5748 | . . . . . . . . 9 | |
26 | oveq 5748 | . . . . . . . . 9 | |
27 | 25, 26 | oveq12d 5760 | . . . . . . . 8 |
28 | 22, 27 | breq12d 3912 | . . . . . . 7 |
29 | 28 | ralbidv 2414 | . . . . . 6 |
30 | 24, 29 | anbi12d 464 | . . . . 5 |
31 | 30 | 2ralbidv 2436 | . . . 4 |
32 | 31 | elrab 2813 | . . 3 |
33 | 21, 32 | syl6bb 195 | . 2 |
34 | sqxpexg 4625 | . . . 4 | |
35 | elmapg 6523 | . . . 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 1316 wcel 1465 wral 2393 crab 2397 cvv 2660 class class class wbr 3899 cxp 4507 wfn 5088 wf 5089 cfv 5093 (class class class)co 5742 cmap 6510 cc0 7588 cxr 7767 cle 7769 cxad 9525 cxmet 12076 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-map 6512 df-pnf 7770 df-mnf 7771 df-xr 7772 df-xmet 12084 |
This theorem is referenced by: isxmetd 12443 xmetf 12446 ismet2 12450 xmeteq0 12455 xmettri2 12457 |
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