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Theorem pconncn 32368
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 32367 . . . 4 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
32simprbi 497 . . 3 (𝐽 ∈ PConn → ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
4 eqeq2 2830 . . . . . 6 (𝑥 = 𝐴 → ((𝑓‘0) = 𝑥 ↔ (𝑓‘0) = 𝐴))
54anbi1d 629 . . . . 5 (𝑥 = 𝐴 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦)))
65rexbidv 3294 . . . 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 2830 . . . . . 6 (𝑦 = 𝐵 → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝐵))
87anbi2d 628 . . . . 5 (𝑦 = 𝐵 → (((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
98rexbidv 3294 . . . 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 3630 . . 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 1108 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 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136   cuni 4830  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526  Topctop 21429   Cn ccn 21760  IIcii 23410  PConncpconn 32363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-pconn 32365
This theorem is referenced by:  cnpconn  32374  pconnconn  32375  txpconn  32376  ptpconn  32377  connpconn  32379  pconnpi1  32381  cvmlift3lem2  32464  cvmlift3lem7  32469
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