MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscnp3 Unicode version

Theorem iscnp3 16990
Description: The predicate " 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 16983 . 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 5407 . . . . . . . . . 10  |-  ( F : X --> Y  ->  Fun  F )
32ad2antlr 707 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  Fun  F )
4 toponss 16683 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
54adantlr 695 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  X )
6 fdm 5409 . . . . . . . . . . 11  |-  ( F : X --> Y  ->  dom  F  =  X )
76ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  dom  F  =  X )
85, 7sseqtr4d 3228 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  dom  F )
9 funimass3 5657 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
103, 8, 9syl2anc 642 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  ( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
1110anbi2d 684 . . . . . . 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 2573 . . . . . 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 307 . . . . 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 2576 . . . 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 622 . . 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 976 . 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 244 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   `'ccnv 4704   dom cdm 4705   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874  TopOnctopon 16648    CnP ccnp 16971
This theorem is referenced by:  cncnpi  17023  cnpdis  17037  cmptdst  25671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-cnp 16974
  Copyright terms: Public domain W3C validator