![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
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 7416 | . . 3 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽)) | |
2 | fveq2 6891 | . . . . 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 3342 | . 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 34208 | . 2 ⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))} | |
7 | 5, 6 | elrab2 3686 | 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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {csn 4628 class class class wbr 5148 × cxp 5674 ‘cfv 6543 (class class class)co 7408 0cc0 11109 1c1 11110 [,]cicc 13326 Cn ccn 22727 IIcii 24390 ≃phcphtpc 24484 PConncpconn 34205 SConncsconn 34206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7411 df-sconn 34208 |
This theorem is referenced by: sconnpconn 34213 sconnpht 34215 sconnpi1 34225 txsconn 34227 cvxsconn 34229 |
Copyright terms: Public domain | W3C validator |