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Theorem ispcon 28305
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 4253 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 ispcon.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2526 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 oveq2 6290 . . . . 5  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
54rexeqdv 3065 . . . 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 3072 . . 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 3072 . 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 28303 . 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 3263 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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   U.cuni 4245   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489   Topctop 19158    Cn ccn 19488   IIcii 21111  PConcpcon 28301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-pcon 28303
This theorem is referenced by:  pconcn  28306  pcontop  28307  cnpcon  28312  txpcon  28314  ptpcon  28315  indispcon  28316  conpcon  28317  cvxpcon  28324
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