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Theorem cnf2 21551
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 21537 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
21simprbda 491 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
323impa 1090 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068  wcel 2048  wral 3082  ccnv 5399  cima 5403  wf 6178  cfv 6182  (class class class)co 6970  TopOnctopon 21212   Cn ccn 21526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-map 8200  df-top 21196  df-topon 21213  df-cn 21529
This theorem is referenced by:  iscncl  21571  cncls2  21575  cncls  21576  cnntr  21577  cnrest2  21588  cnrest2r  21589  ptcn  21929  txdis1cn  21937  lmcn2  21951  cnmpt11  21965  cnmpt1t  21967  cnmpt12  21969  cnmpt21  21973  cnmpt2t  21975  cnmpt22  21976  cnmpt22f  21977  cnmptcom  21980  cnmptkp  21982  cnmptk1  21983  cnmpt1k  21984  cnmptkk  21985  cnmptk1p  21987  cnmptk2  21988  cnmpt2k  21990  qtopss  22017  qtopeu  22018  qtopomap  22020  qtopcmap  22021  hmeof1o2  22065  xpstopnlem1  22111  xkocnv  22116  xkohmeo  22117  qtophmeo  22119  cnmpt1plusg  22389  cnmpt2plusg  22390  tsmsmhm  22447  cnmpt1vsca  22495  cnmpt2vsca  22496  cnmpt1ds  23143  cnmpt2ds  23144  fsumcn  23171  cnmpopc  23225  htpyco1  23275  htpyco2  23276  phtpyco2  23287  pi1xfrf  23350  pi1xfr  23352  pi1xfrcnvlem  23353  pi1xfrcnv  23354  pi1cof  23356  pi1coghm  23358  cnmpt1ip  23543  cnmpt2ip  23544  txsconnlem  32032  txsconn  32033  cvmlift3lem6  32116  fcnre  40645  refsumcn  40650  refsum2cnlem1  40657  fprodcnlem  41257  icccncfext  41546  itgsubsticclem  41636
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