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Theorem cnprcl 17301
Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
iscnp2.1  |-  X  = 
U. J
Assertion
Ref Expression
cnprcl  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  P  e.  X )

Proof of Theorem cnprcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscnp2.1 . . . 4  |-  X  = 
U. J
2 eqid 2435 . . . 4  |-  U. K  =  U. K
31, 2iscnp2 17295 . . 3  |-  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( ( J  e.  Top  /\  K  e.  Top  /\  P  e.  X )  /\  ( F : X --> 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.  X ) )
54simp3d 971 1  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  P  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   U.cuni 4007   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073   Topctop 16950    CnP ccnp 17281
This theorem is referenced by:  cnprcl2  17307  cnpco  17323  cnprest2  17346  ghmcnp  18136  metcnpi  18566  metcnpi2  18567  metcnpi3  18568  limccnp  19770  limccnp2  19771
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-top 16955  df-topon 16958  df-cnp 17284
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