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Theorem pconcn 23755
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 23754 . . . 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 2292 . . . . . 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 2564 . . . 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 2292 . . . . . 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 2564 . . . 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 2890 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   U.cuni 3827   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738   Topctop 16631    Cn ccn 16954   IIcii 18379  PConcpcon 23750
This theorem is referenced by:  cnpcon  23761  pconcon  23762  txpcon  23763  ptpcon  23764  conpcon  23766  pconpi1  23768  cvmlift3lem2  23851  cvmlift3lem7  23856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-pcon 23752
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