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Mirrors > Home > ILE Home > Th. List > restopnb | Unicode version |
Description: If is an open subset of the subspace base set , then any subset of is open iff it is open in . (Contributed by Mario Carneiro, 2-Mar-2015.) |
Ref | Expression |
---|---|
restopnb | ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr3 989 | . . . . . . 7 | |
2 | simpr2 988 | . . . . . . 7 | |
3 | 1, 2 | sstrd 3102 | . . . . . 6 |
4 | df-ss 3079 | . . . . . 6 | |
5 | 3, 4 | sylib 121 | . . . . 5 |
6 | 5 | eqcomd 2143 | . . . 4 |
7 | ineq1 3265 | . . . . . 6 | |
8 | 7 | rspceeqv 2802 | . . . . 5 |
9 | 8 | expcom 115 | . . . 4 |
10 | 6, 9 | syl 14 | . . 3 |
11 | inass 3281 | . . . . . 6 | |
12 | simprr 521 | . . . . . . . 8 | |
13 | 12 | ineq1d 3271 | . . . . . . 7 |
14 | simplr3 1025 | . . . . . . . . 9 | |
15 | df-ss 3079 | . . . . . . . . 9 | |
16 | 14, 15 | sylib 121 | . . . . . . . 8 |
17 | 16 | adantrr 470 | . . . . . . 7 |
18 | 13, 17 | eqtr3d 2172 | . . . . . 6 |
19 | simplr2 1024 | . . . . . . . . 9 | |
20 | sseqin2 3290 | . . . . . . . . 9 | |
21 | 19, 20 | sylib 121 | . . . . . . . 8 |
22 | 21 | ineq2d 3272 | . . . . . . 7 |
23 | 22 | adantrr 470 | . . . . . 6 |
24 | 11, 18, 23 | 3eqtr3a 2194 | . . . . 5 |
25 | simplll 522 | . . . . . 6 | |
26 | simprl 520 | . . . . . 6 | |
27 | simplr1 1023 | . . . . . 6 | |
28 | inopn 12159 | . . . . . 6 | |
29 | 25, 26, 27, 28 | syl3anc 1216 | . . . . 5 |
30 | 24, 29 | eqeltrd 2214 | . . . 4 |
31 | 30 | rexlimdvaa 2548 | . . 3 |
32 | 10, 31 | impbid 128 | . 2 |
33 | elrest 12116 | . . 3 ↾t | |
34 | 33 | adantr 274 | . 2 ↾t |
35 | 32, 34 | bitr4d 190 | 1 ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wrex 2415 cin 3065 wss 3066 (class class class)co 5767 ↾t crest 12109 ctop 12153 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-rest 12111 df-top 12154 |
This theorem is referenced by: restopn2 12341 |
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