Theorem cnf 22305

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 22297	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
4	3	simprbi 496	. 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
5	4	simpld 494	1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∪ cuni 4836  ^◡ccnv 5579   “ cima 5583  ⟶wf 6414  (class class class)co 7255  Topctop 21950   Cn ccn 22283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-top 21951  df-topon 21968  df-cn 22286

This theorem is referenced by:  cnco  22325  cnclima  22327  cnntri  22330  cnclsi  22331  cnss1  22335  cnss2  22336  cncnpi  22337  cncnp2  22340  cnrest  22344  cnrest2  22345  cnt0  22405  cnt1  22409  cnhaus  22413  dnsconst  22437  cncmp  22451  rncmp  22455  imacmp  22456  cnconn  22481  connima  22484  conncn  22485  2ndcomap  22517  kgencn2  22616  kgencn3  22617  txcnmpt  22683  uptx  22684  txcn  22685  hauseqlcld  22705  xkohaus  22712  xkoptsub  22713  xkopjcn  22715  xkoco1cn  22716  xkoco2cn  22717  xkococnlem  22718  cnmpt11f  22723  cnmpt21f  22731  hmeocnv  22821  hmeores  22830  txhmeo  22862  cnextfres  23128  bndth  24027  evth  24028  evth2  24029  htpyco2  24048  phtpyco2  24059  reparphti  24066  copco  24087  pcopt  24091  pcopt2  24092  pcoass  24093  pcorevlem  24095  pcorev2  24097  hauseqcn  31750  pl1cn  31807  rrhf  31848  esumcocn  31948  cnmbfm  32130  cnpconn  33092  ptpconn  33095  sconnpi1  33101  txsconnlem  33102  cvxsconn  33105  cvmseu  33138  cvmopnlem  33140  cvmfolem  33141  cvmliftmolem1  33143  cvmliftmolem2  33144  cvmliftlem3  33149  cvmliftlem6  33152  cvmliftlem7  33153  cvmliftlem8  33154  cvmliftlem9  33155  cvmliftlem10  33156  cvmliftlem11  33157  cvmliftlem13  33158  cvmliftlem15  33160  cvmlift2lem3  33167  cvmlift2lem5  33169  cvmlift2lem7  33171  cvmlift2lem9  33173  cvmlift2lem10  33174  cvmliftphtlem  33179  cvmlift3lem1  33181  cvmlift3lem2  33182  cvmlift3lem4  33184  cvmlift3lem5  33185  cvmlift3lem6  33186  cvmlift3lem7  33187  cvmlift3lem8  33188  cvmlift3lem9  33189  poimirlem31  35735  poimir  35737  broucube  35738  cnres2  35848  cnresima  35849  hausgraph  40953  refsum2cnlem1  42469  itgsubsticclem  43406  stoweidlem62  43493  cnfsmf  44163  cnneiima  46098  sepfsepc  46109