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Theorem cnf2 22400
Description: A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnf2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)

Proof of Theorem cnf2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iscn 22386 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
21simprbda 499 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
323impa 1109 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086  wcel 2106  wral 3064  ccnv 5588  cima 5592  wf 6429  cfv 6433  (class class class)co 7275  TopOnctopon 22059   Cn ccn 22375
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
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 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-top 22043  df-topon 22060  df-cn 22378
This theorem is referenced by:  iscncl  22420  cncls2  22424  cncls  22425  cnntr  22426  cnrest2  22437  cnrest2r  22438  ptcn  22778  txdis1cn  22786  lmcn2  22800  cnmpt11  22814  cnmpt1t  22816  cnmpt12  22818  cnmpt21  22822  cnmpt2t  22824  cnmpt22  22825  cnmpt22f  22826  cnmptcom  22829  cnmptkp  22831  cnmptk1  22832  cnmpt1k  22833  cnmptkk  22834  cnmptk1p  22836  cnmptk2  22837  cnmpt2k  22839  qtopss  22866  qtopeu  22867  qtopomap  22869  qtopcmap  22870  hmeof1o2  22914  xpstopnlem1  22960  xkocnv  22965  xkohmeo  22966  qtophmeo  22968  cnmpt1plusg  23238  cnmpt2plusg  23239  tsmsmhm  23297  cnmpt1vsca  23345  cnmpt2vsca  23346  cnmpt1ds  24005  cnmpt2ds  24006  fsumcn  24033  cnmpopc  24091  htpyco1  24141  htpyco2  24142  phtpyco2  24153  pi1xfrf  24216  pi1xfr  24218  pi1xfrcnvlem  24219  pi1xfrcnv  24220  pi1cof  24222  pi1coghm  24224  cnmpt1ip  24411  cnmpt2ip  24412  txsconnlem  33202  txsconn  33203  cvmlift3lem6  33286  fcnre  42568  refsumcn  42573  refsum2cnlem1  42580  fprodcnlem  43140  icccncfext  43428  itgsubsticclem  43516
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