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Mirrors > Home > MPE Home > Th. List > Mathboxes > issconn | Structured version Visualization version GIF version |
Description: The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
issconn | ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7439 | . . 3 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽)) | |
2 | fveq2 6907 | . . . . 5 ⊢ (𝑗 = 𝐽 → ( ≃ph‘𝑗) = ( ≃ph‘𝐽)) | |
3 | 2 | breqd 5159 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}) ↔ 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))) |
4 | 3 | imbi2d 340 | . . 3 ⊢ (𝑗 = 𝐽 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) |
5 | 1, 4 | raleqbidv 3344 | . 2 ⊢ (𝑗 = 𝐽 → (∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) |
6 | df-sconn 35207 | . 2 ⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))} | |
7 | 5, 6 | elrab2 3698 | 1 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {csn 4631 class class class wbr 5148 × cxp 5687 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 [,]cicc 13387 Cn ccn 23248 IIcii 24915 ≃phcphtpc 25015 PConncpconn 35204 SConncsconn 35205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-sconn 35207 |
This theorem is referenced by: sconnpconn 35212 sconnpht 35214 sconnpi1 35224 txsconn 35226 cvxsconn 35228 |
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