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Theorem iscnp3 19854
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 19847 . 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 5658 . . . . . . . . . 10  |-  ( F : X --> Y  ->  Fun  F )
32ad2antlr 724 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  Fun  F )
4 toponss 19538 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
54adantlr 712 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  X )
6 fdm 5660 . . . . . . . . . . 11  |-  ( F : X --> Y  ->  dom  F  =  X )
76ad2antlr 724 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  dom  F  =  X )
85, 7sseqtr4d 3471 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  dom  F )
9 funimass3 5922 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
103, 8, 9syl2anc 659 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  ( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
1110anbi2d 701 . . . . . . 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 2907 . . . . . 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 314 . . . . 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 2835 . . . 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 639 . . 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 1015 . 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 253 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 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   A.wral 2746   E.wrex 2747    C_ wss 3406   `'ccnv 4929   dom cdm 4930   "cima 4933   Fun wfun 5507   -->wf 5509   ` cfv 5513  (class class class)co 6218  TopOnctopon 19503    CnP ccnp 19835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-map 7362  df-top 19507  df-topon 19510  df-cnp 19838
This theorem is referenced by:  cncnpi  19888  cnpdis  19903
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