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Mirrors > Home > ILE Home > Th. List > restopn2 | Unicode version |
Description: If is open, then is open in iff it is an open subset of . (Contributed by Mario Carneiro, 2-Mar-2015.) |
Ref | Expression |
---|---|
restopn2 | ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 3811 | . . . . 5 ↾t ↾t | |
2 | elssuni 3811 | . . . . . . 7 | |
3 | eqid 2164 | . . . . . . . 8 | |
4 | 3 | restuni 12719 | . . . . . . 7 ↾t |
5 | 2, 4 | sylan2 284 | . . . . . 6 ↾t |
6 | 5 | sseq2d 3167 | . . . . 5 ↾t |
7 | 1, 6 | syl5ibr 155 | . . . 4 ↾t |
8 | 7 | pm4.71rd 392 | . . 3 ↾t ↾t |
9 | simpll 519 | . . . . 5 | |
10 | simplr 520 | . . . . 5 | |
11 | ssidd 3158 | . . . . 5 | |
12 | simpr 109 | . . . . 5 | |
13 | restopnb 12728 | . . . . 5 ↾t | |
14 | 9, 10, 10, 11, 12, 13 | syl23anc 1234 | . . . 4 ↾t |
15 | 14 | pm5.32da 448 | . . 3 ↾t |
16 | 8, 15 | bitr4d 190 | . 2 ↾t |
17 | ancom 264 | . 2 | |
18 | 16, 17 | bitrdi 195 | 1 ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wss 3111 cuni 3783 (class class class)co 5836 ↾t crest 12498 ctop 12542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-rest 12500 df-topgen 12519 df-top 12543 df-topon 12556 df-bases 12588 |
This theorem is referenced by: restdis 12731 |
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