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Theorem cnprcl 17233
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 2389 . . . 4  |-  U. K  =  U. K
31, 2iscnp2 17227 . . 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 1649    e. wcel 1717   A.wral 2651   E.wrex 2652    C_ wss 3265   U.cuni 3959   "cima 4823   -->wf 5392   ` cfv 5396  (class class class)co 6022   Topctop 16883    CnP ccnp 17213
This theorem is referenced by:  cnprcl2  17239  cnpco  17255  cnprest2  17278  ghmcnp  18067  metcnpi  18466  metcnpi2  18467  metcnpi3  18468  limccnp  19647  limccnp2  19648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-map 6958  df-top 16888  df-topon 16891  df-cnp 17216
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