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Theorem ispconn 33757
Description: The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
ispconn.1 𝑋 = 𝐽
Assertion
Ref Expression
ispconn (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem ispconn
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 unieq 4876 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ispconn.1 . . . 4 𝑋 = 𝐽
31, 2eqtr4di 2794 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 oveq2 7364 . . . . 5 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
54rexeqdv 3314 . . . 4 (𝑗 = 𝐽 → (∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
63, 5raleqbidv 3319 . . 3 (𝑗 = 𝐽 → (∀𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
73, 6raleqbidv 3319 . 2 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
8 df-pconn 33755 . 2 PConn = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
97, 8elrab2 3648 1 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073   cuni 4865  cfv 6496  (class class class)co 7356  0cc0 11050  1c1 11051  Topctop 22240   Cn ccn 22573  IIcii 24236  PConncpconn 33753
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 2707
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 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7359  df-pconn 33755
This theorem is referenced by:  pconncn  33758  pconntop  33759  cnpconn  33764  txpconn  33766  ptpconn  33767  indispconn  33768  connpconn  33769  cvxpconn  33776
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