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Theorem cnf 20310
Description: A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
iscnp2.1  |-  X  = 
U. J
iscnp2.2  |-  Y  = 
U. K
Assertion
Ref Expression
cnf  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )

Proof of Theorem cnf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iscnp2.1 . . . 4  |-  X  = 
U. J
2 iscnp2.2 . . . 4  |-  Y  = 
U. K
31, 2iscn2 20302 . . 3  |-  ( F  e.  ( J  Cn  K )  <->  ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : X --> Y  /\  A. x  e.  K  ( `' F " x )  e.  J ) ) )
43simprbi 470 . 2  |-  ( F  e.  ( J  Cn  K )  ->  ( F : X --> Y  /\  A. x  e.  K  ( `' F " x )  e.  J ) )
54simpld 465 1  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748   U.cuni 4211   `'ccnv 4851   "cima 4855   -->wf 5596  (class class class)co 6314   Topctop 19965    Cn ccn 20288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-map 7499  df-top 19969  df-topon 19971  df-cn 20291
This theorem is referenced by:  cnco  20330  cnclima  20332  cnntri  20335  cnclsi  20336  cnss1  20340  cnss2  20341  cncnpi  20342  cncnp2  20345  cnrest  20349  cnrest2  20350  cnt0  20410  cnt1  20414  cnhaus  20418  dnsconst  20442  cncmp  20455  rncmp  20459  imacmp  20460  cnconn  20485  conima  20488  concn  20489  2ndcomap  20521  kgencn2  20620  kgencn3  20621  txcnmpt  20687  uptx  20688  txcn  20689  hauseqlcld  20709  xkohaus  20716  xkoptsub  20717  xkopjcn  20719  xkoco1cn  20720  xkoco2cn  20721  xkococnlem  20722  cnmpt11f  20727  cnmpt21f  20735  hmeocnv  20825  hmeores  20834  txhmeo  20866  cnextfres  21132  bndth  22034  evth  22035  evth2  22036  htpyco2  22058  phtpyco2  22069  reparphti  22076  copco  22097  pcopt  22101  pcopt2  22102  pcoass  22103  pcorevlem  22105  pcorev2  22107  hauseqcn  28749  pl1cn  28809  rrhf  28850  esumcocn  28949  cnmbfm  29133  cnpcon  30001  ptpcon  30004  sconpi1  30010  txsconlem  30011  cvxscon  30014  cvmseu  30047  cvmopnlem  30049  cvmfolem  30050  cvmliftmolem1  30052  cvmliftmolem2  30053  cvmliftlem3  30058  cvmliftlem6  30061  cvmliftlem7  30062  cvmliftlem8  30063  cvmliftlem9  30064  cvmliftlem10  30065  cvmliftlem11  30066  cvmliftlem13  30067  cvmliftlem15  30069  cvmlift2lem3  30076  cvmlift2lem5  30078  cvmlift2lem7  30080  cvmlift2lem9  30082  cvmlift2lem10  30083  cvmliftphtlem  30088  cvmlift3lem1  30090  cvmlift3lem2  30091  cvmlift3lem4  30093  cvmlift3lem5  30094  cvmlift3lem6  30095  cvmlift3lem7  30096  cvmlift3lem8  30097  cvmlift3lem9  30098  poimirlem31  32015  poimir  32017  broucube  32018  cnres2  32139  cnresima  32140  hausgraph  36133  refsum2cnlem1  37397  itgsubsticclem  37889  stoweidlem62  37960  stoweidlem62OLD  37961
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