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Theorem issconn 34217
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 7417 . . 3 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2 fveq2 6892 . . . . 5 (𝑗 = 𝐽 → ( ≃ph𝑗) = ( ≃ph𝐽))
32breqd 5160 . . . 4 (𝑗 = 𝐽 → (𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}) ↔ 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})))
43imbi2d 341 . . 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 3343 . 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 34213 . 2 SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}))}
75, 6elrab2 3687 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 205  wa 397   = wceq 1542  wcel 2107  wral 3062  {csn 4629   class class class wbr 5149   × cxp 5675  cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  [,]cicc 13327   Cn ccn 22728  IIcii 24391  phcphtpc 24485  PConncpconn 34210  SConncsconn 34211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-sconn 34213
This theorem is referenced by:  sconnpconn  34218  sconnpht  34220  sconnpi1  34230  txsconn  34232  cvxsconn  34234
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