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Theorem pconcn 24911
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
pconcn  |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  E. f  e.  ( 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 pconcn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispcon.1 . . . . 5  |-  X  = 
U. J
21ispcon 24910 . . . 4  |-  ( 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 ) ) )
32simprbi 451 . . 3  |-  ( J  e. PCon  ->  A. x  e.  X  A. y  e.  X  E. f  e.  (
II  Cn  J )
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y ) )
4 eqeq2 2445 . . . . . 6  |-  ( x  =  A  ->  (
( f `  0
)  =  x  <->  ( f `  0 )  =  A ) )
54anbi1d 686 . . . . 5  |-  ( x  =  A  ->  (
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y )  <->  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  y ) ) )
65rexbidv 2726 . . . 4  |-  ( x  =  A  ->  ( E. f  e.  (
II  Cn  J )
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y )  <->  E. f  e.  ( II  Cn  J ) ( ( f ` 
0 )  =  A  /\  ( f ` 
1 )  =  y ) ) )
7 eqeq2 2445 . . . . . 6  |-  ( y  =  B  ->  (
( f `  1
)  =  y  <->  ( f `  1 )  =  B ) )
87anbi2d 685 . . . . 5  |-  ( y  =  B  ->  (
( ( f ` 
0 )  =  A  /\  ( f ` 
1 )  =  y )  <->  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )
98rexbidv 2726 . . . 4  |-  ( y  =  B  ->  ( E. f  e.  (
II  Cn  J )
( ( f ` 
0 )  =  A  /\  ( f ` 
1 )  =  y )  <->  E. f  e.  ( II  Cn  J ) ( ( f ` 
0 )  =  A  /\  ( f ` 
1 )  =  B ) ) )
106, 9rspc2v 3058 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  E. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  x  /\  ( f `
 1 )  =  y )  ->  E. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )
113, 10syl5com 28 . 2  |-  ( J  e. PCon  ->  ( ( A  e.  X  /\  B  e.  X )  ->  E. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )
12113impib 1151 1  |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  E. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   U.cuni 4015   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991   Topctop 16958    Cn ccn 17288   IIcii 18905  PConcpcon 24906
This theorem is referenced by:  cnpcon  24917  pconcon  24918  txpcon  24919  ptpcon  24920  conpcon  24922  pconpi1  24924  cvmlift3lem2  25007  cvmlift3lem7  25012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-pcon 24908
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