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Theorem cnf 21782
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 = 𝐽
iscnp2.2 𝑌 = 𝐾
Assertion
Ref Expression
cnf (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌)

Proof of Theorem cnf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iscnp2.1 . . . 4 𝑋 = 𝐽
2 iscnp2.2 . . . 4 𝑌 = 𝐾
31, 2iscn2 21774 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
43simprbi 497 . 2 (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽))
54simpld 495 1 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135   cuni 4830  ccnv 5547  cima 5551  wf 6344  (class class class)co 7145  Topctop 21429   Cn ccn 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-top 21430  df-topon 21447  df-cn 21763
This theorem is referenced by:  cnco  21802  cnclima  21804  cnntri  21807  cnclsi  21808  cnss1  21812  cnss2  21813  cncnpi  21814  cncnp2  21817  cnrest  21821  cnrest2  21822  cnt0  21882  cnt1  21886  cnhaus  21890  dnsconst  21914  cncmp  21928  rncmp  21932  imacmp  21933  cnconn  21958  connima  21961  conncn  21962  2ndcomap  21994  kgencn2  22093  kgencn3  22094  txcnmpt  22160  uptx  22161  txcn  22162  hauseqlcld  22182  xkohaus  22189  xkoptsub  22190  xkopjcn  22192  xkoco1cn  22193  xkoco2cn  22194  xkococnlem  22195  cnmpt11f  22200  cnmpt21f  22208  hmeocnv  22298  hmeores  22307  txhmeo  22339  cnextfres  22605  bndth  23489  evth  23490  evth2  23491  htpyco2  23510  phtpyco2  23521  reparphti  23528  copco  23549  pcopt  23553  pcopt2  23554  pcoass  23555  pcorevlem  23557  pcorev2  23559  hauseqcn  31037  pl1cn  31097  rrhf  31138  esumcocn  31238  cnmbfm  31420  cnpconn  32374  ptpconn  32377  sconnpi1  32383  txsconnlem  32384  cvxsconn  32387  cvmseu  32420  cvmopnlem  32422  cvmfolem  32423  cvmliftmolem1  32425  cvmliftmolem2  32426  cvmliftlem3  32431  cvmliftlem6  32434  cvmliftlem7  32435  cvmliftlem8  32436  cvmliftlem9  32437  cvmliftlem10  32438  cvmliftlem11  32439  cvmliftlem13  32440  cvmliftlem15  32442  cvmlift2lem3  32449  cvmlift2lem5  32451  cvmlift2lem7  32453  cvmlift2lem9  32455  cvmlift2lem10  32456  cvmliftphtlem  32461  cvmlift3lem1  32463  cvmlift3lem2  32464  cvmlift3lem4  32466  cvmlift3lem5  32467  cvmlift3lem6  32468  cvmlift3lem7  32469  cvmlift3lem8  32470  cvmlift3lem9  32471  poimirlem31  34804  poimir  34806  broucube  34807  cnres2  34922  cnresima  34923  hausgraph  39690  refsum2cnlem1  41171  itgsubsticclem  42136  stoweidlem62  42224  cnfsmf  42894
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