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Theorem cnptop1 19504
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 2462 . . . 4  |-  U. J  =  U. J
2 eqid 2462 . . . 4  |-  U. K  =  U. K
31, 2iscnp2 19501 . . 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 460 . 2  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  ( J  e.  Top  /\  K  e.  Top  /\  P  e. 
U. J ) )
54simp1d 1003 1  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    e. wcel 1762   A.wral 2809   E.wrex 2810    C_ wss 3471   U.cuni 4240   "cima 4997   -->wf 5577   ` cfv 5581  (class class class)co 6277   Topctop 19156    CnP ccnp 19487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-map 7414  df-top 19161  df-topon 19164  df-cnp 19490
This theorem is referenced by:  cnpco  19529  cncnp2  19543  cnpresti  19550  cnprest2  19552  lmcnp  19566
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