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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 32931 | . 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∀wral 3064 {csn 4557 class class class wbr 5069 × cxp 5566 ‘cfv 6400 (class class class)co 7234 0cc0 10758 1c1 10759 [,]cicc 12967 Cn ccn 22152 IIcii 23803 ≃phcphtpc 23897 PConncpconn 32924 SConncsconn 32925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ral 3069 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-iota 6358 df-fv 6408 df-ov 7237 df-sconn 32927 |
This theorem is referenced by: sconntop 32933 txsconn 32946 resconn 32951 iinllyconn 32959 cvmlift2lem10 33017 cvmlift3lem2 33025 cvmlift3 33033 |
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