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Theorem cnpimaex 17320
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 2436 . . . . . 6  |-  U. J  =  U. J
2 eqid 2436 . . . . . 6  |-  U. K  =  U. K
31, 2iscnp2 17303 . . . . 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 451 . . . 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 450 . . 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 2497 . . . . 5  |-  ( y  =  A  ->  (
( F `  P
)  e.  y  <->  ( F `  P )  e.  A
) )
7 sseq2 3370 . . . . . . 7  |-  ( y  =  A  ->  (
( F " x
)  C_  y  <->  ( F " x )  C_  A
) )
87anbi2d 685 . . . . . 6  |-  ( y  =  A  ->  (
( P  e.  x  /\  ( F " x
)  C_  y )  <->  ( P  e.  x  /\  ( F " x ) 
C_  A ) ) )
98rexbidv 2726 . . . . 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 312 . . . 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 3049 . . 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 16 . 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 1147 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   U.cuni 4015   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081   Topctop 16958    CnP ccnp 17289
This theorem is referenced by:  iscnp4  17327  cnpnei  17328  cnpco  17331  cncnp  17344  cnpresti  17352  lmcnp  17368  txcnpi  17640  txcnp  17652  ptcnplem  17653  cnpflfi  18031  ghmcnp  18144  xrlimcnp  20807  cnambfre  26255
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-top 16963  df-topon 16966  df-cnp 17292
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