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Theorem cnpimaex 18858
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 2441 . . . . . 6  |-  U. J  =  U. J
2 eqid 2441 . . . . . 6  |-  U. K  =  U. K
31, 2iscnp2 18841 . . . . 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 464 . . . 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 463 . . 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 2502 . . . . 5  |-  ( y  =  A  ->  (
( F `  P
)  e.  y  <->  ( F `  P )  e.  A
) )
7 sseq2 3376 . . . . . . 7  |-  ( y  =  A  ->  (
( F " x
)  C_  y  <->  ( F " x )  C_  A
) )
87anbi2d 703 . . . . . 6  |-  ( y  =  A  ->  (
( P  e.  x  /\  ( F " x
)  C_  y )  <->  ( P  e.  x  /\  ( F " x ) 
C_  A ) ) )
98rexbidv 2734 . . . . 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 320 . . . 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 3068 . . 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 1181 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714    C_ wss 3326   U.cuni 4089   "cima 4841   -->wf 5412   ` cfv 5416  (class class class)co 6089   Topctop 18496    CnP ccnp 18827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-map 7214  df-top 18501  df-topon 18504  df-cnp 18830
This theorem is referenced by:  iscnp4  18865  cnpnei  18866  cnpco  18869  cncnp  18882  cnpresti  18890  lmcnp  18906  txcnpi  19179  txcnp  19191  ptcnplem  19192  cnpflfi  19570  ghmcnp  19683  xrlimcnp  22360  cnambfre  28437
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