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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 6886 | . . 3 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽)) | |
2 | fveq2 6411 | . . . . 5 ⊢ (𝑗 = 𝐽 → ( ≃ph‘𝑗) = ( ≃ph‘𝐽)) | |
3 | 2 | breqd 4854 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}) ↔ 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))) |
4 | 3 | imbi2d 332 | . . 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 3335 | . 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 31721 | . 2 ⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))} | |
7 | 5, 6 | elrab2 3560 | 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 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 {csn 4368 class class class wbr 4843 × cxp 5310 ‘cfv 6101 (class class class)co 6878 0cc0 10224 1c1 10225 [,]cicc 12427 Cn ccn 21357 IIcii 23006 ≃phcphtpc 23096 PConncpconn 31718 SConncsconn 31719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-iota 6064 df-fv 6109 df-ov 6881 df-sconn 31721 |
This theorem is referenced by: sconnpconn 31726 sconnpht 31728 sconnpi1 31738 txsconn 31740 cvxsconn 31742 |
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