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Mirrors > Home > ILE Home > Th. List > resttopon2 | GIF version |
Description: The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
resttopon2 | β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβ(π΄ β© π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 13785 | . . 3 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | resttop 13941 | . . 3 β’ ((π½ β Top β§ π΄ β π) β (π½ βΎt π΄) β Top) | |
3 | 1, 2 | sylan 283 | . 2 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt π΄) β Top) |
4 | toponuni 13786 | . . . . 5 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
5 | 4 | ineq2d 3348 | . . . 4 β’ (π½ β (TopOnβπ) β (π΄ β© π) = (π΄ β© βͺ π½)) |
6 | 5 | adantr 276 | . . 3 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π΄ β© π) = (π΄ β© βͺ π½)) |
7 | eqid 2187 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
8 | 7 | restuni2 13948 | . . . 4 β’ ((π½ β Top β§ π΄ β π) β (π΄ β© βͺ π½) = βͺ (π½ βΎt π΄)) |
9 | 1, 8 | sylan 283 | . . 3 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π΄ β© βͺ π½) = βͺ (π½ βΎt π΄)) |
10 | 6, 9 | eqtrd 2220 | . 2 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π΄ β© π) = βͺ (π½ βΎt π΄)) |
11 | istopon 13784 | . 2 β’ ((π½ βΎt π΄) β (TopOnβ(π΄ β© π)) β ((π½ βΎt π΄) β Top β§ (π΄ β© π) = βͺ (π½ βΎt π΄))) | |
12 | 3, 10, 11 | sylanbrc 417 | 1 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβ(π΄ β© π))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1363 β wcel 2158 β© cin 3140 βͺ cuni 3821 βcfv 5228 (class class class)co 5888 βΎt crest 12705 Topctop 13768 TopOnctopon 13781 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-rest 12707 df-topgen 12726 df-top 13769 df-topon 13782 df-bases 13814 |
This theorem is referenced by: lmss 14017 |
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