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Mirrors > Home > MPE Home > Th. List > Mathboxes > sconnpconn | Structured version Visualization version GIF version |
Description: A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
sconnpconn | ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issconn 32370 | . 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {csn 4557 class class class wbr 5057 × cxp 5546 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 [,]cicc 12729 Cn ccn 21760 IIcii 23410 ≃phcphtpc 23500 PConncpconn 32363 SConncsconn 32364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-sconn 32366 |
This theorem is referenced by: sconntop 32372 txsconn 32385 resconn 32390 iinllyconn 32398 cvmlift2lem10 32456 cvmlift3lem2 32464 cvmlift3 32472 |
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