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Theorem issconn 35231
Description: The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
issconn (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
Distinct variable group:   𝑓,𝐽

Proof of Theorem issconn
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7439 . . 3 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2 fveq2 6906 . . . . 5 (𝑗 = 𝐽 → ( ≃ph𝑗) = ( ≃ph𝐽))
32breqd 5154 . . . 4 (𝑗 = 𝐽 → (𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}) ↔ 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})))
43imbi2d 340 . . 3 (𝑗 = 𝐽 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
51, 4raleqbidv 3346 . 2 (𝑗 = 𝐽 → (∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
6 df-sconn 35227 . 2 SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}))}
75, 6elrab2 3695 1 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {csn 4626   class class class wbr 5143   × cxp 5683  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  [,]cicc 13390   Cn ccn 23232  IIcii 24901  phcphtpc 25001  PConncpconn 35224  SConncsconn 35225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-sconn 35227
This theorem is referenced by:  sconnpconn  35232  sconnpht  35234  sconnpi1  35244  txsconn  35246  cvxsconn  35248
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