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Theorem cnptop2 20194
Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnptop2  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  K  e.  Top )

Proof of Theorem cnptop2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2422 . . . 4  |-  U. J  =  U. J
2 eqid 2422 . . . 4  |-  U. K  =  U. K
31, 2iscnp2 20190 . . 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 461 . 2  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  ( J  e.  Top  /\  K  e.  Top  /\  P  e. 
U. J ) )
54simp2d 1018 1  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  K  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1872   A.wral 2708   E.wrex 2709    C_ wss 3372   U.cuni 4155   "cima 4792   -->wf 5533   ` cfv 5537  (class class class)co 6242   Topctop 19852    CnP ccnp 20176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-fv 5545  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-1st 6744  df-2nd 6745  df-map 7422  df-top 19856  df-topon 19858  df-cnp 20179
This theorem is referenced by:  cnpco  20218  cncnp2  20232  cnpresti  20239  cnprest  20240  lmcnp  20255  cnpflfi  20949  flfcnp  20954  flfcnp2  20957
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