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Mirrors > Home > MPE Home > Th. List > Mathboxes > istotbnd | Structured version Visualization version Unicode version |
Description: The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
istotbnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6221 | . 2 | |
2 | elfvex 6221 | . . 3 | |
3 | 2 | adantr 481 | . 2 |
4 | fveq2 6191 | . . . . . 6 | |
5 | eqeq2 2633 | . . . . . . . . 9 | |
6 | rexeq 3139 | . . . . . . . . . 10 | |
7 | 6 | ralbidv 2986 | . . . . . . . . 9 |
8 | 5, 7 | anbi12d 747 | . . . . . . . 8 |
9 | 8 | rexbidv 3052 | . . . . . . 7 |
10 | 9 | ralbidv 2986 | . . . . . 6 |
11 | 4, 10 | rabeqbidv 3195 | . . . . 5 |
12 | df-totbnd 33567 | . . . . 5 | |
13 | fvex 6201 | . . . . . 6 | |
14 | 13 | rabex 4813 | . . . . 5 |
15 | 11, 12, 14 | fvmpt 6282 | . . . 4 |
16 | 15 | eleq2d 2687 | . . 3 |
17 | fveq2 6191 | . . . . . . . . . . 11 | |
18 | 17 | oveqd 6667 | . . . . . . . . . 10 |
19 | 18 | eqeq2d 2632 | . . . . . . . . 9 |
20 | 19 | rexbidv 3052 | . . . . . . . 8 |
21 | 20 | ralbidv 2986 | . . . . . . 7 |
22 | 21 | anbi2d 740 | . . . . . 6 |
23 | 22 | rexbidv 3052 | . . . . 5 |
24 | 23 | ralbidv 2986 | . . . 4 |
25 | 24 | elrab 3363 | . . 3 |
26 | 16, 25 | syl6bb 276 | . 2 |
27 | 1, 3, 26 | pm5.21nii 368 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wrex 2913 crab 2916 cvv 3200 cuni 4436 cfv 5888 (class class class)co 6650 cfn 7955 crp 11832 cme 19732 cbl 19733 ctotbnd 33565 |
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 |
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-ne 2795 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-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-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-totbnd 33567 |
This theorem is referenced by: istotbnd2 33569 istotbnd3 33570 totbndmet 33571 totbndss 33576 heibor1 33609 heibor 33620 |
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