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Theorem iscnp3 13336
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 13332 . 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 5363 . . . . . . . . . 10  |-  ( F : X --> Y  ->  Fun  F )
32ad2antlr 489 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  Fun  F )
4 toponss 13157 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
54adantlr 477 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  X )
6 fdm 5366 . . . . . . . . . . 11  |-  ( F : X --> Y  ->  dom  F  =  X )
76ad2antlr 489 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  dom  F  =  X )
85, 7sseqtrrd 3194 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  x  C_  dom  F )
9 funimass3 5627 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
103, 8, 9syl2anc 411 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  F : X --> Y )  /\  x  e.  J )  ->  ( ( F "
x )  C_  y  <->  x 
C_  ( `' F " y ) ) )
1110anbi2d 464 . . . . . . 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 2474 . . . . . 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 230 . . . . 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 2477 . . . 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 452 . . 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 1018 . 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 188 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456    C_ wss 3129   `'ccnv 4621   dom cdm 4622   "cima 4625   Fun wfun 5205   -->wf 5207   ` cfv 5211  (class class class)co 5868  TopOnctopon 13141    CnP ccnp 13319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-map 6643  df-top 13129  df-topon 13142  df-cnp 13322
This theorem is referenced by:  cncnpi  13361  cnpdis  13375
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