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Theorem cnf 19541
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 19533 . . 3  |-  ( F  e.  ( J  Cn  K )  <->  ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : X --> Y  /\  A. x  e.  K  ( `' F " x )  e.  J ) ) )
43simprbi 464 . 2  |-  ( F  e.  ( J  Cn  K )  ->  ( F : X --> Y  /\  A. x  e.  K  ( `' F " x )  e.  J ) )
54simpld 459 1  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   U.cuni 4245   `'ccnv 4998   "cima 5002   -->wf 5584  (class class class)co 6284   Topctop 19189    Cn ccn 19519
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 6576
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-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-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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-top 19194  df-topon 19197  df-cn 19522
This theorem is referenced by:  cnco  19561  cnclima  19563  cnntri  19566  cnclsi  19567  cnss1  19571  cnss2  19572  cncnpi  19573  cncnp2  19576  cnrest  19580  cnrest2  19581  cnt0  19641  cnt1  19645  cnhaus  19649  dnsconst  19673  cncmp  19686  rncmp  19690  imacmp  19691  cnconn  19717  conima  19720  concn  19721  2ndcomap  19753  kgencn2  19821  kgencn3  19822  txcnmpt  19888  uptx  19889  txcn  19890  hauseqlcld  19910  xkohaus  19917  xkoptsub  19918  xkopjcn  19920  xkoco1cn  19921  xkoco2cn  19922  xkococnlem  19923  cnmpt11f  19928  cnmpt21f  19936  hmeocnv  20026  hmeores  20035  txhmeo  20067  bndth  21221  evth  21222  evth2  21223  htpyco2  21242  phtpyco2  21253  reparphti  21260  copco  21281  pcopt  21285  pcopt2  21286  pcoass  21287  pcorevlem  21289  pcorev2  21291  hauseqcn  27541  pl1cn  27601  rrhf  27643  esumcocn  27754  cnmbfm  27902  cnpcon  28343  ptpcon  28346  sconpi1  28352  txsconlem  28353  cvxscon  28356  cvmseu  28389  cvmopnlem  28391  cvmfolem  28392  cvmliftmolem1  28394  cvmliftmolem2  28395  cvmliftlem3  28400  cvmliftlem6  28403  cvmliftlem7  28404  cvmliftlem8  28405  cvmliftlem9  28406  cvmliftlem10  28407  cvmliftlem11  28408  cvmliftlem13  28409  cvmliftlem15  28411  cvmlift2lem3  28418  cvmlift2lem5  28420  cvmlift2lem7  28422  cvmlift2lem9  28424  cvmlift2lem10  28425  cvmliftphtlem  28430  cvmlift3lem1  28432  cvmlift3lem2  28433  cvmlift3lem4  28435  cvmlift3lem5  28436  cvmlift3lem6  28437  cvmlift3lem7  28438  cvmlift3lem8  28439  cvmlift3lem9  28440  cnres2  29890  cnresima  29891  hausgraph  30805  refsum2cnlem1  31018  itgsubsticclem  31321  stoweidlem62  31390
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