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Theorem cncnp2 18844
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 18803 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2 cncnp.1 . . . . . 6  |-  X  = 
U. J
32toptopon 18497 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
41, 3sylib 196 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
5 cntop2 18804 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
6 cncnp.2 . . . . . 6  |-  Y  = 
U. K
76toptopon 18497 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
85, 7sylib 196 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  (TopOn `  Y )
)
92, 6cnf 18809 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
104, 8, 9jca31 531 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
1110adantl 463 . 2  |-  ( ( X  =/=  (/)  /\  F  e.  ( J  Cn  K
) )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
12 r19.2z 3766 . . 3  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )  ->  E. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )
13 cnptop1 18805 . . . . . 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 18806 . . . . . 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 18810 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  F : X --> Y )
1814, 16, 17jca31 531 . . . 4  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
1918rexlimivw 2835 . . 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 18843 . . 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 895 . 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 888 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 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   (/)c0 3634   U.cuni 4088   -->wf 5411   ` cfv 5415  (class class class)co 6090   Topctop 18457  TopOnctopon 18458    Cn ccn 18787    CnP ccnp 18788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7212  df-topgen 14378  df-top 18462  df-topon 18465  df-cn 18790  df-cnp 18791
This theorem is referenced by: (None)
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