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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 4205 |
. . . . 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 4165 |
. . 3
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5 | simpl 109 |
. . 3
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6 | restval 12830 |
. . 3
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7 | 4, 5, 6 | syl2anc 411 |
. 2
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8 | 3 | sselda 3175 |
. . . . . . . 8
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9 | 8 | elpwid 3608 |
. . . . . . 7
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10 | df-ss 3162 |
. . . . . . 7
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11 | 9, 10 | sylib 122 |
. . . . . 6
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12 | 11 | mpteq2dva 4115 |
. . . . 5
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13 | mptresid 4986 |
. . . . 5
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14 | 12, 13 | eqtrdi 2238 |
. . . 4
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15 | 14 | rneqd 4881 |
. . 3
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16 | rnresi 5010 |
. . 3
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17 | 15, 16 | eqtrdi 2238 |
. 2
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18 | 7, 17 | eqtrd 2222 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-id 4318 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-ov 5909 df-oprab 5910 df-mpo 5911 df-rest 12826 |
This theorem is referenced by: restid 12835 topnidg 12837 |
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