Theorem cnf 21848

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 21840	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
4	3	simprbi 499	. 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
5	4	simpld 497	1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∪ cuni 4832  ^◡ccnv 5549   “ cima 5553  ⟶wf 6346  (class class class)co 7150  Topctop 21495   Cn ccn 21826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8402  df-top 21496  df-topon 21513  df-cn 21829

This theorem is referenced by:  cnco  21868  cnclima  21870  cnntri  21873  cnclsi  21874  cnss1  21878  cnss2  21879  cncnpi  21880  cncnp2  21883  cnrest  21887  cnrest2  21888  cnt0  21948  cnt1  21952  cnhaus  21956  dnsconst  21980  cncmp  21994  rncmp  21998  imacmp  21999  cnconn  22024  connima  22027  conncn  22028  2ndcomap  22060  kgencn2  22159  kgencn3  22160  txcnmpt  22226  uptx  22227  txcn  22228  hauseqlcld  22248  xkohaus  22255  xkoptsub  22256  xkopjcn  22258  xkoco1cn  22259  xkoco2cn  22260  xkococnlem  22261  cnmpt11f  22266  cnmpt21f  22274  hmeocnv  22364  hmeores  22373  txhmeo  22405  cnextfres  22671  bndth  23556  evth  23557  evth2  23558  htpyco2  23577  phtpyco2  23588  reparphti  23595  copco  23616  pcopt  23620  pcopt2  23621  pcoass  23622  pcorevlem  23624  pcorev2  23626  hauseqcn  31133  pl1cn  31193  rrhf  31234  esumcocn  31334  cnmbfm  31516  cnpconn  32472  ptpconn  32475  sconnpi1  32481  txsconnlem  32482  cvxsconn  32485  cvmseu  32518  cvmopnlem  32520  cvmfolem  32521  cvmliftmolem1  32523  cvmliftmolem2  32524  cvmliftlem3  32529  cvmliftlem6  32532  cvmliftlem7  32533  cvmliftlem8  32534  cvmliftlem9  32535  cvmliftlem10  32536  cvmliftlem11  32537  cvmliftlem13  32538  cvmliftlem15  32540  cvmlift2lem3  32547  cvmlift2lem5  32549  cvmlift2lem7  32551  cvmlift2lem9  32553  cvmlift2lem10  32554  cvmliftphtlem  32559  cvmlift3lem1  32561  cvmlift3lem2  32562  cvmlift3lem4  32564  cvmlift3lem5  32565  cvmlift3lem6  32566  cvmlift3lem7  32567  cvmlift3lem8  32568  cvmlift3lem9  32569  poimirlem31  34917  poimir  34919  broucube  34920  cnres2  35035  cnresima  35036  hausgraph  39805  refsum2cnlem1  41287  itgsubsticclem  42252  stoweidlem62  42340  cnfsmf  43010