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Theorem cnptop1 17306
Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnptop1  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  J  e.  Top )

Proof of Theorem cnptop1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . 4  |-  U. J  =  U. J
2 eqid 2436 . . . 4  |-  U. K  =  U. K
31, 2iscnp2 17303 . . 3  |-  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( ( J  e.  Top  /\  K  e.  Top  /\  P  e. 
U. J )  /\  ( F : U. J --> U. K  /\  A. y  e.  K  ( ( F `  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) ) ) ) )
43simplbi 447 . 2  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  ( J  e.  Top  /\  K  e.  Top  /\  P  e. 
U. J ) )
54simp1d 969 1  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   U.cuni 4015   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081   Topctop 16958    CnP ccnp 17289
This theorem is referenced by:  cnpco  17331  cncnp2  17345  cnpresti  17352  cnprest2  17354  lmcnp  17368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-top 16963  df-topon 16966  df-cnp 17292
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