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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sconpcon | Structured version Visualization version GIF version |
Description: A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
sconpcon | ⊢ (𝐽 ∈ SCon → 𝐽 ∈ PCon) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isscon 30462 | . 2 ⊢ (𝐽 ∈ SCon ↔ (𝐽 ∈ PCon ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | |
2 | 1 | simplbi 475 | 1 ⊢ (𝐽 ∈ SCon → 𝐽 ∈ PCon) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {csn 4125 class class class wbr 4583 × cxp 5036 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 [,]cicc 12049 Cn ccn 20838 IIcii 22486 ≃phcphtpc 22576 PConcpcon 30455 SConcscon 30456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-scon 30458 |
This theorem is referenced by: scontop 30464 txscon 30477 rescon 30482 iinllycon 30490 cvmlift2lem10 30548 cvmlift3lem2 30556 cvmlift3 30564 |
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