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Mirrors > Home > ILE Home > Th. List > setsmstsetg | Unicode version |
Description: The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.) |
Ref | Expression |
---|---|
setsms.x |
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setsms.d |
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setsms.k |
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setsmsbasg.m |
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setsmsbasg.d |
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Ref | Expression |
---|---|
setsmstsetg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsmsbasg.m |
. . 3
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2 | setsmsbasg.d |
. . 3
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3 | tsetslid 12634 |
. . . 4
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4 | 3 | setsslid 12504 |
. . 3
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5 | 1, 2, 4 | syl2anc 411 |
. 2
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6 | setsms.k |
. . 3
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7 | 6 | fveq2d 5517 |
. 2
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8 | 5, 7 | eqtr4d 2213 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7898 ax-resscn 7899 ax-1re 7901 ax-addrcl 7904 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-iota 5176 df-fun 5216 df-fv 5222 df-ov 5874 df-oprab 5875 df-mpo 5876 df-inn 8915 df-2 8973 df-3 8974 df-4 8975 df-5 8976 df-6 8977 df-7 8978 df-8 8979 df-9 8980 df-ndx 12456 df-slot 12457 df-sets 12460 df-tset 12546 |
This theorem is referenced by: (None) |
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