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Theorem pconcn 27277
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 27276 . . . 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 464 . . 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 2469 . . . . . 6  |-  ( x  =  A  ->  (
( f `  0
)  =  x  <->  ( f `  0 )  =  A ) )
54anbi1d 704 . . . . 5  |-  ( x  =  A  ->  (
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y )  <->  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  y ) ) )
65rexbidv 2868 . . . 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 2469 . . . . . 6  |-  ( y  =  B  ->  (
( f `  1
)  =  y  <->  ( f `  1 )  =  B ) )
87anbi2d 703 . . . . 5  |-  ( y  =  B  ->  (
( ( f ` 
0 )  =  A  /\  ( f ` 
1 )  =  y )  <->  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )
98rexbidv 2868 . . . 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 3186 . . 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 30 . 2  |-  ( J  e. PCon  ->  ( ( A  e.  X  /\  B  e.  X )  ->  E. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )
12113impib 1186 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   U.cuni 4202   ` cfv 5529  (class class class)co 6203   0cc0 9396   1c1 9397   Topctop 18633    Cn ccn 18963   IIcii 20586  PConcpcon 27272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206  df-pcon 27274
This theorem is referenced by:  cnpcon  27283  pconcon  27284  txpcon  27285  ptpcon  27286  conpcon  27288  pconpi1  27290  cvmlift3lem2  27373  cvmlift3lem7  27378
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