Theorem cnf 22406

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:𝑋⟶𝑌)

Proof of Theorem cnf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscnp2.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2		iscnp2.2	. . . 4 ⊢ 𝑌 = ∪ 𝐾
3	1, 2	iscn2 22398	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
4	3	simprbi 497	. 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
5	4	simpld 495	1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2107  ∀wral 3065  ∪ cuni 4840  ^◡ccnv 5589   “ cima 5593  ⟶wf 6433  (class class class)co 7284  Topctop 22051   Cn ccn 22384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-sbc 3718  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-fv 6445  df-ov 7287  df-oprab 7288  df-mpo 7289  df-map 8626  df-top 22052  df-topon 22069  df-cn 22387

This theorem is referenced by:  cnco  22426  cnclima  22428  cnntri  22431  cnclsi  22432  cnss1  22436  cnss2  22437  cncnpi  22438  cncnp2  22441  cnrest  22445  cnrest2  22446  cnt0  22506  cnt1  22510  cnhaus  22514  dnsconst  22538  cncmp  22552  rncmp  22556  imacmp  22557  cnconn  22582  connima  22585  conncn  22586  2ndcomap  22618  kgencn2  22717  kgencn3  22718  txcnmpt  22784  uptx  22785  txcn  22786  hauseqlcld  22806  xkohaus  22813  xkoptsub  22814  xkopjcn  22816  xkoco1cn  22817  xkoco2cn  22818  xkococnlem  22819  cnmpt11f  22824  cnmpt21f  22832  hmeocnv  22922  hmeores  22931  txhmeo  22963  cnextfres  23229  bndth  24130  evth  24131  evth2  24132  htpyco2  24151  phtpyco2  24162  reparphti  24169  copco  24190  pcopt  24194  pcopt2  24195  pcoass  24196  pcorevlem  24198  pcorev2  24200  hauseqcn  31857  pl1cn  31914  rrhf  31957  esumcocn  32057  cnmbfm  32239  cnpconn  33201  ptpconn  33204  sconnpi1  33210  txsconnlem  33211  cvxsconn  33214  cvmseu  33247  cvmopnlem  33249  cvmfolem  33250  cvmliftmolem1  33252  cvmliftmolem2  33253  cvmliftlem3  33258  cvmliftlem6  33261  cvmliftlem7  33262  cvmliftlem8  33263  cvmliftlem9  33264  cvmliftlem10  33265  cvmliftlem11  33266  cvmliftlem13  33267  cvmliftlem15  33269  cvmlift2lem3  33276  cvmlift2lem5  33278  cvmlift2lem7  33280  cvmlift2lem9  33282  cvmlift2lem10  33283  cvmliftphtlem  33288  cvmlift3lem1  33290  cvmlift3lem2  33291  cvmlift3lem4  33293  cvmlift3lem5  33294  cvmlift3lem6  33295  cvmlift3lem7  33296  cvmlift3lem8  33297  cvmlift3lem9  33298  poimirlem31  35817  poimir  35819  broucube  35820  cnres2  35930  cnresima  35931  hausgraph  41044  refsum2cnlem1  42587  itgsubsticclem  43523  stoweidlem62  43610  cnfsmf  44285  cnneiima  46221  sepfsepc  46232