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Theorem cnprcl 19505
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 2460 . . . 4  |-  U. K  =  U. K
31, 2iscnp2 19499 . . 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 460 . 2  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  ( J  e.  Top  /\  K  e.  Top  /\  P  e.  X ) )
54simp3d 1005 1  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  P  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    C_ wss 3469   U.cuni 4238   "cima 4995   -->wf 5575   ` cfv 5579  (class class class)co 6275   Topctop 19154    CnP ccnp 19485
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 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
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 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-top 19159  df-topon 19162  df-cnp 19488
This theorem is referenced by:  cnprcl2  19511  cnpco  19527  cnprest2  19550  ghmcnp  20341  metcnpi  20775  metcnpi2  20776  metcnpi3  20777  limccnp  22023  limccnp2  22024  fouriercnp  31482
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