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Theorem pconncn 35229
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
pconncn ((𝐽 ∈ PConn ∧ 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 pconncn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispconn.1 . . . . 5 𝑋 = 𝐽
21ispconn 35228 . . . 4 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
32simprbi 496 . . 3 (𝐽 ∈ PConn → ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
4 eqeq2 2749 . . . . . 6 (𝑥 = 𝐴 → ((𝑓‘0) = 𝑥 ↔ (𝑓‘0) = 𝐴))
54anbi1d 631 . . . . 5 (𝑥 = 𝐴 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦)))
65rexbidv 3179 . . . 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 2749 . . . . . 6 (𝑦 = 𝐵 → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝐵))
87anbi2d 630 . . . . 5 (𝑦 = 𝐵 → (((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
98rexbidv 3179 . . . 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 3633 . . 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 ∈ PConn → ((𝐴𝑋𝐵𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
12113impib 1117 1 ((𝐽 ∈ PConn ∧ 𝐴𝑋𝐵𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070   cuni 4907  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  Topctop 22899   Cn ccn 23232  IIcii 24901  PConncpconn 35224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-pconn 35226
This theorem is referenced by:  cnpconn  35235  pconnconn  35236  txpconn  35237  ptpconn  35238  connpconn  35240  pconnpi1  35242  cvmlift3lem2  35325  cvmlift3lem7  35330
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