Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pconcn Structured version   Visualization version   GIF version

Theorem pconcn 30460
 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
pconcn ((𝐽 ∈ PCon ∧ 𝐴𝑋𝐵𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝐽
Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem pconcn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispcon.1 . . . . 5 𝑋 = 𝐽
21ispcon 30459 . . . 4 (𝐽 ∈ PCon ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
32simprbi 479 . . 3 (𝐽 ∈ PCon → ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
4 eqeq2 2621 . . . . . 6 (𝑥 = 𝐴 → ((𝑓‘0) = 𝑥 ↔ (𝑓‘0) = 𝐴))
54anbi1d 737 . . . . 5 (𝑥 = 𝐴 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦)))
65rexbidv 3034 . . . 4 (𝑥 = 𝐴 → (∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦)))
7 eqeq2 2621 . . . . . 6 (𝑦 = 𝐵 → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝐵))
87anbi2d 736 . . . . 5 (𝑦 = 𝐵 → (((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
98rexbidv 3034 . . . 4 (𝑦 = 𝐵 → (∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
106, 9rspc2v 3293 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
113, 10syl5com 31 . 2 (𝐽 ∈ PCon → ((𝐴𝑋𝐵𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
12113impib 1254 1 ((𝐽 ∈ PCon ∧ 𝐴𝑋𝐵𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = 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:  cnpcon  30466  pconcon  30467  txpcon  30468  ptpcon  30469  conpcon  30471  pconpi1  30473  cvmlift3lem2  30556  cvmlift3lem7  30561
 Copyright terms: Public domain W3C validator