ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iscnp3 Unicode version

Theorem iscnp3 12402
Description: The predicate "the class  F is a continuous function from topology  J to topology  K at point  P". (Contributed by NM, 15-May-2007.)
Assertion
Ref Expression
iscnp3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
( F : X --> Y  /\  A. y  e.  K  ( ( F `
 P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  x  C_  ( `' F "
y ) ) ) ) ) )
Distinct variable groups:    x, y, F   
x, J, y    x, K, y    x, X, y   
x, Y, y    x, P, y

Proof of Theorem iscnp3
StepHypRef Expression
1 iscnp 12398 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
( F : X --> Y  /\  A. y  e.  K  ( ( F `
 P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) ) ) ) )
2 ffun 5279 . . . . . . . . . 10  |-  ( F : X --> Y  ->  Fun  F )
32ad2antlr 481 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  Fun  F )
4 toponss 12223 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
54adantlr 469 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  X )
6 fdm 5282 . . . . . . . . . . 11  |-  ( F : X --> Y  ->  dom  F  =  X )
76ad2antlr 481 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  dom  F  =  X )
85, 7sseqtrrd 3137 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  dom  F )
9 funimass3 5540 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
103, 8, 9syl2anc 409 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  ( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
1110anbi2d 460 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  ( ( P  e.  x  /\  ( F
" x )  C_  y )  <->  ( P  e.  x  /\  x  C_  ( `' F "
y ) ) ) )
1211rexbidva 2435 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  -> 
( E. x  e.  J  ( P  e.  x  /\  ( F
" x )  C_  y )  <->  E. x  e.  J  ( P  e.  x  /\  x  C_  ( `' F "
y ) ) ) )
1312imbi2d 229 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  -> 
( ( ( F `
 P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) )  <->  ( ( F `  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  x  C_  ( `' F "
y ) ) ) ) )
1413ralbidv 2438 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  -> 
( A. y  e.  K  ( ( F `
 P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) )  <->  A. y  e.  K  ( ( F `  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  x  C_  ( `' F "
y ) ) ) ) )
1514pm5.32da 448 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( ( F : X --> Y  /\  A. y  e.  K  ( ( F `  P
)  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  y )
) )  <->  ( F : X --> Y  /\  A. y  e.  K  (
( F `  P
)  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  x  C_  ( `' F " y ) ) ) ) ) )
16153ad2ant1 1003 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  ->  ( ( F : X --> Y  /\  A. y  e.  K  ( ( F `  P
)  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  y )
) )  <->  ( F : X --> Y  /\  A. y  e.  K  (
( F `  P
)  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  x  C_  ( `' F " y ) ) ) ) ) )
171, 16bitrd 187 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
( F : X --> Y  /\  A. y  e.  K  ( ( F `
 P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  x  C_  ( `' F "
y ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418    C_ wss 3072   `'ccnv 4542   dom cdm 4543   "cima 4546   Fun wfun 5121   -->wf 5123   ` cfv 5127  (class class class)co 5778  TopOnctopon 12207    CnP ccnp 12385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-map 6548  df-top 12195  df-topon 12208  df-cnp 12388
This theorem is referenced by:  cncnpi  12427  cnpdis  12441
  Copyright terms: Public domain W3C validator