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Mirrors > Home > ILE Home > Th. List > restid2 | Unicode version |
Description: The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
restid2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4178 |
. . . . 5
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2 | 1 | adantr 276 |
. . . 4
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3 | simpr 110 |
. . . 4
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4 | 2, 3 | ssexd 4141 |
. . 3
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5 | simpl 109 |
. . 3
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6 | restval 12680 |
. . 3
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7 | 4, 5, 6 | syl2anc 411 |
. 2
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8 | 3 | sselda 3155 |
. . . . . . . 8
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9 | 8 | elpwid 3586 |
. . . . . . 7
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10 | df-ss 3142 |
. . . . . . 7
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11 | 9, 10 | sylib 122 |
. . . . . 6
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12 | 11 | mpteq2dva 4091 |
. . . . 5
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13 | mptresid 4958 |
. . . . 5
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14 | 12, 13 | eqtrdi 2226 |
. . . 4
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15 | 14 | rneqd 4853 |
. . 3
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16 | rnresi 4982 |
. . 3
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17 | 15, 16 | eqtrdi 2226 |
. 2
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18 | 7, 17 | eqtrd 2210 |
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-coll 4116 ax-sep 4119 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 |
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-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-iun 3887 df-br 4002 df-opab 4063 df-mpt 4064 df-id 4291 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-f1 5218 df-fo 5219 df-f1o 5220 df-fv 5221 df-ov 5873 df-oprab 5874 df-mpo 5875 df-rest 12676 |
This theorem is referenced by: restid 12685 topnidg 12687 |
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