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Mirrors > Home > MPE Home > Th. List > isxmet | Structured version Visualization version 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 3212 | . . . . 5 | |
2 | xpeq12 5134 | . . . . . . . . 9 | |
3 | 2 | anidms 677 | . . . . . . . 8 |
4 | 3 | oveq2d 6666 | . . . . . . 7 |
5 | raleq 3138 | . . . . . . . . . 10 | |
6 | 5 | anbi2d 740 | . . . . . . . . 9 |
7 | 6 | raleqbi1dv 3146 | . . . . . . . 8 |
8 | 7 | raleqbi1dv 3146 | . . . . . . 7 |
9 | 4, 8 | rabeqbidv 3195 | . . . . . 6 |
10 | df-xmet 19739 | . . . . . 6 | |
11 | ovex 6678 | . . . . . . 7 | |
12 | 11 | rabex 4813 | . . . . . 6 |
13 | 9, 10, 12 | fvmpt 6282 | . . . . 5 |
14 | 1, 13 | syl 17 | . . . 4 |
15 | 14 | eleq2d 2687 | . . 3 |
16 | oveq 6656 | . . . . . . . 8 | |
17 | 16 | eqeq1d 2624 | . . . . . . 7 |
18 | 17 | bibi1d 333 | . . . . . 6 |
19 | oveq 6656 | . . . . . . . . 9 | |
20 | oveq 6656 | . . . . . . . . 9 | |
21 | 19, 20 | oveq12d 6668 | . . . . . . . 8 |
22 | 16, 21 | breq12d 4666 | . . . . . . 7 |
23 | 22 | ralbidv 2986 | . . . . . 6 |
24 | 18, 23 | anbi12d 747 | . . . . 5 |
25 | 24 | 2ralbidv 2989 | . . . 4 |
26 | 25 | elrab 3363 | . . 3 |
27 | 15, 26 | syl6bb 276 | . 2 |
28 | xrex 11829 | . . . 4 | |
29 | sqxpexg 6963 | . . . 4 | |
30 | elmapg 7870 | . . . 4 | |
31 | 28, 29, 30 | sylancr 695 | . . 3 |
32 | 31 | anbi1d 741 | . 2 |
33 | 27, 32 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 crab 2916 cvv 3200 class class class wbr 4653 cxp 5112 wf 5884 cfv 5888 (class class class)co 6650 cmap 7857 cc0 9936 cxr 10073 cle 10075 cxad 11944 cxmt 19731 |
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-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-xr 10078 df-xmet 19739 |
This theorem is referenced by: isxmetd 22131 xmetf 22134 ismet2 22138 xmeteq0 22143 xmettri2 22145 imasf1oxmet 22180 pstmxmet 29940 |
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