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Theorem pconcn 24039
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 24038 . . . 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 450 . . 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 2325 . . . . . 6  |-  ( x  =  A  ->  (
( f `  0
)  =  x  <->  ( f `  0 )  =  A ) )
54anbi1d 685 . . . . 5  |-  ( x  =  A  ->  (
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y )  <->  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  y ) ) )
65rexbidv 2598 . . . 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 2325 . . . . . 6  |-  ( y  =  B  ->  (
( f `  1
)  =  y  <->  ( f `  1 )  =  B ) )
87anbi2d 684 . . . . 5  |-  ( y  =  B  ->  (
( ( f ` 
0 )  =  A  /\  ( f ` 
1 )  =  y )  <->  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )
98rexbidv 2598 . . . 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 2924 . . 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 26 . 2  |-  ( J  e. PCon  ->  ( ( A  e.  X  /\  B  e.  X )  ->  E. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )
12113impib 1149 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 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   E.wrex 2578   U.cuni 3864   ` cfv 5292  (class class class)co 5900   0cc0 8782   1c1 8783   Topctop 16687    Cn ccn 17010   IIcii 18431  PConcpcon 24034
This theorem is referenced by:  cnpcon  24045  pconcon  24046  txpcon  24047  ptpcon  24048  conpcon  24050  pconpi1  24052  cvmlift3lem2  24135  cvmlift3lem7  24140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903  df-pcon 24036
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