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Theorem cnf2 21860
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 21846 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
21simprbda 502 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
323impa 1107 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2115  wral 3133  ccnv 5542  cima 5546  wf 6340  cfv 6344  (class class class)co 7150  TopOnctopon 21521   Cn ccn 21835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8405  df-top 21505  df-topon 21522  df-cn 21838
This theorem is referenced by:  iscncl  21880  cncls2  21884  cncls  21885  cnntr  21886  cnrest2  21897  cnrest2r  21898  ptcn  22238  txdis1cn  22246  lmcn2  22260  cnmpt11  22274  cnmpt1t  22276  cnmpt12  22278  cnmpt21  22282  cnmpt2t  22284  cnmpt22  22285  cnmpt22f  22286  cnmptcom  22289  cnmptkp  22291  cnmptk1  22292  cnmpt1k  22293  cnmptkk  22294  cnmptk1p  22296  cnmptk2  22297  cnmpt2k  22299  qtopss  22326  qtopeu  22327  qtopomap  22329  qtopcmap  22330  hmeof1o2  22374  xpstopnlem1  22420  xkocnv  22425  xkohmeo  22426  qtophmeo  22428  cnmpt1plusg  22698  cnmpt2plusg  22699  tsmsmhm  22757  cnmpt1vsca  22805  cnmpt2vsca  22806  cnmpt1ds  23453  cnmpt2ds  23454  fsumcn  23481  cnmpopc  23539  htpyco1  23589  htpyco2  23590  phtpyco2  23601  pi1xfrf  23664  pi1xfr  23666  pi1xfrcnvlem  23667  pi1xfrcnv  23668  pi1cof  23670  pi1coghm  23672  cnmpt1ip  23857  cnmpt2ip  23858  txsconnlem  32547  txsconn  32548  cvmlift3lem6  32631  fcnre  41575  refsumcn  41580  refsum2cnlem1  41587  fprodcnlem  42168  icccncfext  42456  itgsubsticclem  42544
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