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Theorem cnfnc 23433
 Description: Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
cnfnc
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem cnfnc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnfn 23385 . . . 4
21simprbi 451 . . 3
3 oveq2 6089 . . . . . . . 8
43fveq2d 5732 . . . . . . 7
54breq1d 4222 . . . . . 6
6 fveq2 5728 . . . . . . . . 9
76oveq2d 6097 . . . . . . . 8
87fveq2d 5732 . . . . . . 7
98breq1d 4222 . . . . . 6
105, 9imbi12d 312 . . . . 5
1110rexralbidv 2749 . . . 4
12 breq2 4216 . . . . . 6
1312imbi2d 308 . . . . 5
1413rexralbidv 2749 . . . 4
1511, 14rspc2v 3058 . . 3
162, 15syl5com 28 . 2
17163impib 1151 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706   class class class wbr 4212  wf 5450  cfv 5454  (class class class)co 6081  cc 8988   clt 9120   cmin 9291  crp 10612  cabs 12039  chil 22422  cno 22426   cmv 22428  ccnfn 22456 This theorem is referenced by:  nmcfnexi  23554 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-13 1727  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-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-hilex 22502 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-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-cnfn 23350
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