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Theorem ispcon 28935
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 4243 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 ispcon.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2513 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 oveq2 6278 . . . . 5  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
54rexeqdv 3058 . . . 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 3065 . . 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 3065 . 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 28933 . 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 3256 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   U.cuni 4235   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482   Topctop 19564    Cn ccn 19895   IIcii 21548  PConcpcon 28931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-pcon 28933
This theorem is referenced by:  pconcn  28936  pcontop  28937  cnpcon  28942  txpcon  28944  ptpcon  28945  indispcon  28946  conpcon  28947  cvxpcon  28954
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