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Theorem cncnp2 17026
Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
cncnp.1  |-  X  = 
U. J
cncnp.2  |-  Y  = 
U. K
Assertion
Ref Expression
cncnp2  |-  ( X  =/=  (/)  ->  ( F  e.  ( J  Cn  K
)  <->  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) )
Distinct variable groups:    x, F    x, J    x, K    x, X    x, Y

Proof of Theorem cncnp2
StepHypRef Expression
1 cntop1 16986 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2 cncnp.1 . . . . . 6  |-  X  = 
U. J
32toptopon 16687 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
41, 3sylib 188 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
5 cntop2 16987 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
6 cncnp.2 . . . . . 6  |-  Y  = 
U. K
76toptopon 16687 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
85, 7sylib 188 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  (TopOn `  Y )
)
92, 6cnf 16992 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
104, 8, 9jca31 520 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
1110adantl 452 . 2  |-  ( ( X  =/=  (/)  /\  F  e.  ( J  Cn  K
) )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
12 r19.2z 3556 . . 3  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )  ->  E. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )
13 cnptop1 16988 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  J  e.  Top )
1413, 3sylib 188 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  J  e.  (TopOn `  X )
)
15 cnptop2 16989 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  K  e.  Top )
1615, 7sylib 188 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  K  e.  (TopOn `  Y )
)
172, 6cnpf 16993 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  F : X --> Y )
1814, 16, 17jca31 520 . . . 4  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
1918rexlimivw 2676 . . 3  |-  ( E. x  e.  X  F  e.  ( ( J  CnP  K ) `  x )  ->  ( ( J  e.  (TopOn `  X
)  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y ) )
2012, 19syl 15 . 2  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )  ->  ( ( J  e.  (TopOn `  X
)  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y ) )
21 cncnp 17025 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) ) )
2221baibd 875 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( F  e.  ( J  Cn  K )  <->  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) )
2311, 20, 22pm5.21nd 868 1  |-  ( X  =/=  (/)  ->  ( F  e.  ( J  Cn  K
)  <->  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   (/)c0 3468   U.cuni 3843   -->wf 5267   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648    Cn ccn 16970    CnP ccnp 16971
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-csb 3095  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-iun 3923  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-1st 6138  df-2nd 6139  df-map 6790  df-topgen 13360  df-top 16652  df-topon 16655  df-cn 16973  df-cnp 16974
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