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Theorem pcontop 23756
Description: A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
pcontop  |-  ( J  e. PCon  ->  J  e.  Top )

Proof of Theorem pcontop
Dummy variables  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  U. J  =  U. J
21ispcon 23754 . 2  |-  ( J  e. PCon 
<->  ( J  e.  Top  /\ 
A. x  e.  U. J A. y  e.  U. J E. f  e.  ( II  Cn  J ) ( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y ) ) )
32simplbi 446 1  |-  ( J  e. PCon  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = 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:  scontop  23759  pconcon  23762  txpcon  23763  ptpcon  23764  qtoppcon  23767  pconpi1  23768  sconpi1  23770  cvxscon  23774
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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