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Theorem cnpimaex 16986
Description: Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)
Assertion
Ref Expression
cnpimaex  |-  ( ( F  e.  ( ( J  CnP  K ) `
 P )  /\  A  e.  K  /\  ( F `  P )  e.  A )  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  A )
)
Distinct variable groups:    x, A    x, F    x, J    x, K    x, P

Proof of Theorem cnpimaex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . . 6  |-  U. J  =  U. J
2 eqid 2283 . . . . . 6  |-  U. K  =  U. K
31, 2iscnp2 16969 . . . . 5  |-  ( 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 ) ) ) ) )
43simprbi 450 . . . 4  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  ( F : U. J --> U. K  /\  A. y  e.  K  ( ( F `  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  y )
) ) )
54simprd 449 . . 3  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  A. y  e.  K  ( ( F `  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) ) )
6 eleq2 2344 . . . . 5  |-  ( y  =  A  ->  (
( F `  P
)  e.  y  <->  ( F `  P )  e.  A
) )
7 sseq2 3200 . . . . . . 7  |-  ( y  =  A  ->  (
( F " x
)  C_  y  <->  ( F " x )  C_  A
) )
87anbi2d 684 . . . . . 6  |-  ( y  =  A  ->  (
( P  e.  x  /\  ( F " x
)  C_  y )  <->  ( P  e.  x  /\  ( F " x ) 
C_  A ) ) )
98rexbidv 2564 . . . . 5  |-  ( y  =  A  ->  ( E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  y )  <->  E. x  e.  J  ( P  e.  x  /\  ( F " x ) 
C_  A ) ) )
106, 9imbi12d 311 . . . 4  |-  ( y  =  A  ->  (
( ( F `  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  y )
)  <->  ( ( F `
 P )  e.  A  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  A ) ) ) )
1110rspccv 2881 . . 3  |-  ( A. y  e.  K  (
( F `  P
)  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  y )
)  ->  ( A  e.  K  ->  ( ( F `  P )  e.  A  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  A ) ) ) )
125, 11syl 15 . 2  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  ( A  e.  K  ->  ( ( F `  P
)  e.  A  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  A )
) ) )
13123imp 1145 1  |-  ( ( F  e.  ( ( J  CnP  K ) `
 P )  /\  A  e.  K  /\  ( F `  P )  e.  A )  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631    CnP ccnp 16955
This theorem is referenced by:  cnpnei  16993  cnpco  16996  cncnp  17009  cnpresti  17016  lmcnp  17032  txcnpi  17302  txcnp  17314  ptcnplem  17315  cnpflfi  17694  ghmcnp  17797  xrlimcnp  20263  iscnp4  24975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cnp 16958
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