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Theorem ispconn 31331
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 4476 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ispconn.1 . . . 4 𝑋 = 𝐽
31, 2syl6eqr 2703 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 oveq2 6698 . . . . 5 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
54rexeqdv 3175 . . . 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 3182 . . 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 3182 . 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 31329 . 2 PConn = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
97, 8elrab2 3399 1 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942   cuni 4468  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975  Topctop 20746   Cn ccn 21076  IIcii 22725  PConncpconn 31327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-pconn 31329
This theorem is referenced by:  pconncn  31332  pconntop  31333  cnpconn  31338  txpconn  31340  ptpconn  31341  indispconn  31342  connpconn  31343  cvxpconn  31350
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