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Theorem ispcon 24912
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 4026 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 ispcon.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2488 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 oveq2 6091 . . . . 5  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
54rexeqdv 2913 . . . 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 2918 . . 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 2918 . 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 24910 . 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 3096 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 set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   U.cuni 4017   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993   Topctop 16960    Cn ccn 17290   IIcii 18907  PConcpcon 24908
This theorem is referenced by:  pconcn  24913  pcontop  24914  cnpcon  24919  txpcon  24921  ptpcon  24922  indispcon  24923  conpcon  24924  cvxpcon  24931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-pcon 24910
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