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Theorem issconn 32083
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 7031 . . 3 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2 fveq2 6545 . . . . 5 (𝑗 = 𝐽 → ( ≃ph𝑗) = ( ≃ph𝐽))
32breqd 4979 . . . 4 (𝑗 = 𝐽 → (𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}) ↔ 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})))
43imbi2d 342 . . 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 3363 . 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 32079 . 2 SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}))}
75, 6elrab2 3624 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 207  wa 396   = wceq 1525  wcel 2083  wral 3107  {csn 4478   class class class wbr 4968   × cxp 5448  cfv 6232  (class class class)co 7023  0cc0 10390  1c1 10391  [,]cicc 12595   Cn ccn 21520  IIcii 23170  phcphtpc 23260  PConncpconn 32076  SConncsconn 32077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-iota 6196  df-fv 6240  df-ov 7026  df-sconn 32079
This theorem is referenced by:  sconnpconn  32084  sconnpht  32086  sconnpi1  32096  txsconn  32098  cvxsconn  32100
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