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Theorem pconcn 29397
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 29396 . . . 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 462 . . 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 2415 . . . . . 6  |-  ( x  =  A  ->  (
( f `  0
)  =  x  <->  ( f `  0 )  =  A ) )
54anbi1d 703 . . . . 5  |-  ( x  =  A  ->  (
( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y )  <->  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  y ) ) )
65rexbidv 2915 . . . 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 2415 . . . . . 6  |-  ( y  =  B  ->  (
( f `  1
)  =  y  <->  ( f `  1 )  =  B ) )
87anbi2d 702 . . . . 5  |-  ( y  =  B  ->  (
( ( f ` 
0 )  =  A  /\  ( f ` 
1 )  =  y )  <->  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )
98rexbidv 2915 . . . 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 3166 . . 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 1193 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840   A.wral 2751   E.wrex 2752   U.cuni 4188   ` cfv 5523  (class class class)co 6232   0cc0 9440   1c1 9441   Topctop 19576    Cn ccn 19908   IIcii 21561  PConcpcon 29392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-iota 5487  df-fv 5531  df-ov 6235  df-pcon 29394
This theorem is referenced by:  cnpcon  29403  pconcon  29404  txpcon  29405  ptpcon  29406  conpcon  29408  pconpi1  29410  cvmlift3lem2  29493  cvmlift3lem7  29498
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