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Theorem ispcon 30459
 Description: The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
ispcon.1 𝑋 = 𝐽
Assertion
Ref Expression
ispcon (𝐽 ∈ PCon ↔ (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 ispcon
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 unieq 4380 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ispcon.1 . . . 4 𝑋 = 𝐽
31, 2syl6eqr 2662 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 oveq2 6557 . . . . 5 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
54rexeqdv 3122 . . . 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 3129 . . 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 3129 . 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-pcon 30457 . 2 PCon = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
97, 8elrab2 3333 1 (𝐽 ∈ PCon ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  Topctop 20517   Cn ccn 20838  IIcii 22486  PConcpcon 30455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-pcon 30457 This theorem is referenced by:  pconcn  30460  pcontop  30461  cnpcon  30466  txpcon  30468  ptpcon  30469  indispcon  30470  conpcon  30471  cvxpcon  30478
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