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Theorem cnf2 22000
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 21986 . . 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 1111 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 1088  wcel 2114  wral 3053  ccnv 5524  cima 5528  wf 6335  cfv 6339  (class class class)co 7170  TopOnctopon 21661   Cn ccn 21975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-top 21645  df-topon 21662  df-cn 21978
This theorem is referenced by:  iscncl  22020  cncls2  22024  cncls  22025  cnntr  22026  cnrest2  22037  cnrest2r  22038  ptcn  22378  txdis1cn  22386  lmcn2  22400  cnmpt11  22414  cnmpt1t  22416  cnmpt12  22418  cnmpt21  22422  cnmpt2t  22424  cnmpt22  22425  cnmpt22f  22426  cnmptcom  22429  cnmptkp  22431  cnmptk1  22432  cnmpt1k  22433  cnmptkk  22434  cnmptk1p  22436  cnmptk2  22437  cnmpt2k  22439  qtopss  22466  qtopeu  22467  qtopomap  22469  qtopcmap  22470  hmeof1o2  22514  xpstopnlem1  22560  xkocnv  22565  xkohmeo  22566  qtophmeo  22568  cnmpt1plusg  22838  cnmpt2plusg  22839  tsmsmhm  22897  cnmpt1vsca  22945  cnmpt2vsca  22946  cnmpt1ds  23594  cnmpt2ds  23595  fsumcn  23622  cnmpopc  23680  htpyco1  23730  htpyco2  23731  phtpyco2  23742  pi1xfrf  23805  pi1xfr  23807  pi1xfrcnvlem  23808  pi1xfrcnv  23809  pi1cof  23811  pi1coghm  23813  cnmpt1ip  23999  cnmpt2ip  24000  txsconnlem  32773  txsconn  32774  cvmlift3lem6  32857  fcnre  42126  refsumcn  42131  refsum2cnlem1  42138  fprodcnlem  42702  icccncfext  42990  itgsubsticclem  43078
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