Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cncfval Structured version   Unicode version

Theorem cncfval 18918
 Description: The value of the continuous complex function operation is the set of continuous functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
Assertion
Ref Expression
cncfval
Distinct variable groups:   ,,,,,   ,,,,,

Proof of Theorem cncfval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 9071 . . 3
21elpw2 4364 . 2
31elpw2 4364 . 2
4 oveq2 6089 . . . 4
5 raleq 2904 . . . . . . 7
65rexbidv 2726 . . . . . 6
76ralbidv 2725 . . . . 5
87raleqbi1dv 2912 . . . 4
94, 8rabeqbidv 2951 . . 3
10 oveq1 6088 . . . 4
11 rabeq 2950 . . . 4
1210, 11syl 16 . . 3
13 df-cncf 18908 . . 3
14 ovex 6106 . . . 4
1514rabex 4354 . . 3
169, 12, 13, 15ovmpt2 6209 . 2
172, 3, 16syl2anbr 467 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2705  wrex 2706  crab 2709   wss 3320  cpw 3799   class class class wbr 4212  cfv 5454  (class class class)co 6081   cmap 7018  cc 8988   clt 9120   cmin 9291  crp 10612  cabs 12039  ccncf 18906 This theorem is referenced by:  elcncf  18919 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-cnex 9046 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-cncf 18908
 Copyright terms: Public domain W3C validator