MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnpimaex Structured version   Unicode version

Theorem cnpimaex 20207
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 2429 . . . . . 6  |-  U. J  =  U. J
2 eqid 2429 . . . . . 6  |-  U. K  =  U. K
31, 2iscnp2 20190 . . . . 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 465 . . . 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 464 . . 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 3492 . . . . . . 7  |-  ( y  =  A  ->  (
( F " x
)  C_  y  <->  ( F " x )  C_  A
) )
87anbi2d 708 . . . . . 6  |-  ( y  =  A  ->  (
( P  e.  x  /\  ( F " x
)  C_  y )  <->  ( P  e.  x  /\  ( F " x ) 
C_  A ) ) )
98rexbidv 2946 . . . . 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 321 . . . 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 3185 . . 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 17 . 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 1199 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783    C_ wss 3442   U.cuni 4222   "cima 4857   -->wf 5597   ` cfv 5601  (class class class)co 6305   Topctop 19852    CnP ccnp 20176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-top 19856  df-topon 19858  df-cnp 20179
This theorem is referenced by:  iscnp4  20214  cnpnei  20215  cnpco  20218  cncnp  20231  cnpresti  20239  lmcnp  20255  txcnpi  20558  txcnp  20570  ptcnplem  20571  cnpflfi  20949  ghmcnp  21064  xrlimcnp  23767  cnambfre  31704
  Copyright terms: Public domain W3C validator