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Theorem pconncn 34513
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 34512 . . . 4 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
32simprbi 495 . . 3 (𝐽 ∈ PConn → ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
4 eqeq2 2742 . . . . . 6 (𝑥 = 𝐴 → ((𝑓‘0) = 𝑥 ↔ (𝑓‘0) = 𝐴))
54anbi1d 628 . . . . 5 (𝑥 = 𝐴 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦)))
65rexbidv 3176 . . . 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 2742 . . . . . 6 (𝑦 = 𝐵 → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝐵))
87anbi2d 627 . . . . 5 (𝑦 = 𝐵 → (((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
98rexbidv 3176 . . . 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 3621 . . 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 1114 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 394  w3a 1085   = wceq 1539  wcel 2104  wral 3059  wrex 3068   cuni 4907  cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113  Topctop 22615   Cn ccn 22948  IIcii 24615  PConncpconn 34508
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-pconn 34510
This theorem is referenced by:  cnpconn  34519  pconnconn  34520  txpconn  34521  ptpconn  34522  connpconn  34524  pconnpi1  34526  cvmlift3lem2  34609  cvmlift3lem7  34614
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