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Theorem iscnp3 20258
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 20251 . 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 5748 . . . . . . . . . 10  |-  ( F : X --> Y  ->  Fun  F )
32ad2antlr 731 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  Fun  F )
4 toponss 19942 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
54adantlr 719 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  X )
6 fdm 5750 . . . . . . . . . . 11  |-  ( F : X --> Y  ->  dom  F  =  X )
76ad2antlr 731 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  dom  F  =  X )
85, 7sseqtr4d 3501 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  dom  F )
9 funimass3 6013 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
103, 8, 9syl2anc 665 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  ( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
1110anbi2d 708 . . . . . . 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 2933 . . . . . 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 317 . . . . 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 2861 . . . 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 645 . . 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 1026 . 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 256 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772    C_ wss 3436   `'ccnv 4852   dom cdm 4853   "cima 4856   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305  TopOnctopon 19916    CnP ccnp 20239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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-map 7485  df-top 19919  df-topon 19921  df-cnp 20242
This theorem is referenced by:  cncnpi  20292  cnpdis  20307
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