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Theorem ispcon 27110
Description: The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
ispcon.1  |-  X  = 
U. J
Assertion
Ref Expression
ispcon  |-  ( J  e. PCon 
<->  ( J  e.  Top  /\ 
A. x  e.  X  A. y  e.  X  E. f  e.  (
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 4097 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 ispcon.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2491 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 oveq2 6097 . . . . 5  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
54rexeqdv 2922 . . . 4  |-  ( j  =  J  ->  ( E. f  e.  (
II  Cn  j )
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y )  <->  E. f  e.  ( II  Cn  J ) ( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y ) ) )
63, 5raleqbidv 2929 . . 3  |-  ( j  =  J  ->  ( A. y  e.  U. j E. f  e.  (
II  Cn  j )
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y )  <->  A. y  e.  X  E. f  e.  (
II  Cn  J )
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y ) ) )
73, 6raleqbidv 2929 . 2  |-  ( j  =  J  ->  ( A. x  e.  U. j A. y  e.  U. j E. f  e.  (
II  Cn  j )
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y )  <->  A. x  e.  X  A. y  e.  X  E. f  e.  (
II  Cn  J )
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y ) ) )
8 df-pcon 27108 . 2  |- PCon  =  {
j  e.  Top  |  A. x  e.  U. j A. y  e.  U. j E. f  e.  (
II  Cn  j )
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y ) }
97, 8elrab2 3117 1  |-  ( J  e. PCon 
<->  ( J  e.  Top  /\ 
A. x  e.  X  A. y  e.  X  E. f  e.  (
II  Cn  J )
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714   U.cuni 4089   ` cfv 5416  (class class class)co 6089   0cc0 9280   1c1 9281   Topctop 18496    Cn ccn 18826   IIcii 20449  PConcpcon 27106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-iota 5379  df-fv 5424  df-ov 6092  df-pcon 27108
This theorem is referenced by:  pconcn  27111  pcontop  27112  cnpcon  27117  txpcon  27119  ptpcon  27120  indispcon  27121  conpcon  27122  cvxpcon  27129
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