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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pconntop | Structured version Visualization version GIF version | ||
| Description: A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| pconntop | ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | 1 | ispconn 35228 | . 2 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) | 
| 3 | 2 | simplbi 497 | 1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∪ cuni 4907 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 Topctop 22899 Cn ccn 23232 IIcii 24901 PConncpconn 35224 | 
| 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-ext 2708 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-pconn 35226 | 
| This theorem is referenced by: sconntop 35233 pconnconn 35236 txpconn 35237 ptpconn 35238 qtoppconn 35241 pconnpi1 35242 sconnpi1 35244 cvxsconn 35248 | 
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