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Theorem cnpimaex 19551
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 2467 . . . . . 6  |-  U. J  =  U. J
2 eqid 2467 . . . . . 6  |-  U. K  =  U. K
31, 2iscnp2 19534 . . . . 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 2540 . . . . 5  |-  ( y  =  A  ->  (
( F `  P
)  e.  y  <->  ( F `  P )  e.  A
) )
7 sseq2 3526 . . . . . . 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 2973 . . . . 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 3211 . . 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 1190 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   U.cuni 4245   "cima 5002   -->wf 5584   ` cfv 5588  (class class class)co 6284   Topctop 19189    CnP ccnp 19520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422  df-top 19194  df-topon 19197  df-cnp 19523
This theorem is referenced by:  iscnp4  19558  cnpnei  19559  cnpco  19562  cncnp  19575  cnpresti  19583  lmcnp  19599  txcnpi  19872  txcnp  19884  ptcnplem  19885  cnpflfi  20263  ghmcnp  20376  xrlimcnp  23054  cnambfre  29668
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