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Theorem cncnp2 19548
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 19507 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2 cncnp.1 . . . . . 6  |-  X  = 
U. J
32toptopon 19201 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
41, 3sylib 196 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
5 cntop2 19508 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
6 cncnp.2 . . . . . 6  |-  Y  = 
U. K
76toptopon 19201 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
85, 7sylib 196 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  (TopOn `  Y )
)
92, 6cnf 19513 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
104, 8, 9jca31 534 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
1110adantl 466 . 2  |-  ( ( X  =/=  (/)  /\  F  e.  ( J  Cn  K
) )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
12 r19.2z 3917 . . 3  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )  ->  E. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )
13 cnptop1 19509 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  J  e.  Top )
1413, 3sylib 196 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  J  e.  (TopOn `  X )
)
15 cnptop2 19510 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  K  e.  Top )
1615, 7sylib 196 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  K  e.  (TopOn `  Y )
)
172, 6cnpf 19514 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  F : X --> Y )
1814, 16, 17jca31 534 . . . 4  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
1918rexlimivw 2952 . . 3  |-  ( E. x  e.  X  F  e.  ( ( J  CnP  K ) `  x )  ->  ( ( J  e.  (TopOn `  X
)  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y ) )
2012, 19syl 16 . 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 19547 . . 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 907 . 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 898 1  |-  ( X  =/=  (/)  ->  ( F  e.  ( J  Cn  K
)  <->  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   (/)c0 3785   U.cuni 4245   -->wf 5582   ` cfv 5586  (class class class)co 6282   Topctop 19161  TopOnctopon 19162    Cn ccn 19491    CnP ccnp 19492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-topgen 14695  df-top 19166  df-topon 19169  df-cn 19494  df-cnp 19495
This theorem is referenced by: (None)
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