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Mirrors > Home > ILE Home > Th. List > restuni | GIF version |
Description: The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
restuni.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
restuni | β’ ((π½ β Top β§ π΄ β π) β π΄ = βͺ (π½ βΎt π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restuni.1 | . . . 4 β’ π = βͺ π½ | |
2 | 1 | toptopon 13378 | . . 3 β’ (π½ β Top β π½ β (TopOnβπ)) |
3 | resttopon 13533 | . . 3 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβπ΄)) | |
4 | 2, 3 | sylanb 284 | . 2 β’ ((π½ β Top β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβπ΄)) |
5 | toponuni 13375 | . 2 β’ ((π½ βΎt π΄) β (TopOnβπ΄) β π΄ = βͺ (π½ βΎt π΄)) | |
6 | 4, 5 | syl 14 | 1 β’ ((π½ β Top β§ π΄ β π) β π΄ = βͺ (π½ βΎt π΄)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 β wss 3129 βͺ cuni 3809 βcfv 5214 (class class class)co 5871 βΎt crest 12675 Topctop 13357 TopOnctopon 13370 |
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 4117 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 |
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 3810 df-iun 3888 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-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-fv 5222 df-ov 5874 df-oprab 5875 df-mpo 5876 df-1st 6137 df-2nd 6138 df-rest 12677 df-topgen 12696 df-top 13358 df-topon 13371 df-bases 13403 |
This theorem is referenced by: restuni2 13539 restopn2 13545 reldvg 14010 |
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