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| Mirrors > Home > MPE Home > Th. List > toptopon | Structured version Visualization version GIF version | ||
| Description: Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
| Ref | Expression |
|---|---|
| toptopon.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| toptopon | ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | toptopon.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | istopon 22918 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
| 3 | 1, 2 | mpbiran2 710 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ Top) |
| 4 | 3 | bicomi 224 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 TopOnctopon 22916 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-topon 22917 |
| This theorem is referenced by: toptopon2 22924 eltpsi 22951 restuni 23170 stoig 23171 restlp 23191 restperf 23192 perfopn 23193 iscn2 23246 iscnp2 23247 cncnpi 23286 cncnp2 23289 cnnei 23290 cnrest 23293 cnpresti 23296 cnprest 23297 cnprest2 23298 paste 23302 t1sep2 23377 sshauslem 23380 1stcelcls 23469 kgenuni 23547 iskgen3 23557 txuni 23600 ptuniconst 23606 txcnmpt 23632 txcn 23634 txindis 23642 ptrescn 23647 txcmpb 23652 xkoptsub 23662 xkofvcn 23692 imasnopn 23698 imasncld 23699 imasncls 23700 qtopcmplem 23715 qtopkgen 23718 hmeof1o 23772 hmeores 23779 hmphindis 23805 cmphaushmeo 23808 txhmeo 23811 ptunhmeo 23816 hausflim 23989 flfneii 24000 hausflf 24005 flimfnfcls 24036 flfcntr 24051 cnextfun 24072 cnextfvval 24073 cnextf 24074 cnextcn 24075 cnextfres1 24076 retopon 24784 evth 24991 evth2 24992 qtophaus 33835 rrhre 34022 pconnconn 35236 connpconn 35240 pconnpi1 35242 sconnpi1 35244 txsconnlem 35245 txsconn 35246 cvmsf1o 35277 cvmliftmolem1 35286 cvmliftlem8 35297 cvmlift2lem9a 35308 cvmlift2lem9 35316 cvmlift2lem11 35318 cvmlift2lem12 35319 cvmliftphtlem 35322 cvmlift3lem6 35329 cvmlift3lem8 35331 cvmlift3lem9 35332 cnres2 37770 cnresima 37771 hausgraph 43217 ntrf2 44137 fcnre 45030 |
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