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Theorem cantnffval 7618
 Description: The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnffval.1
cantnffval.2
cantnffval.3
Assertion
Ref Expression
cantnffval CNF OrdIso seq𝜔
Distinct variable groups:   ,,,,,   ,,,,,   ,
Allowed substitution hints:   (,,,,)   (,,,)

Proof of Theorem cantnffval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnffval.2 . 2
2 cantnffval.3 . 2
3 oveq12 6090 . . . . . 6
4 rabeq 2950 . . . . . 6
53, 4syl 16 . . . . 5
6 cantnffval.1 . . . . 5
75, 6syl6eqr 2486 . . . 4
8 simp1l 981 . . . . . . . . . . 11
98oveq1d 6096 . . . . . . . . . 10
109oveq1d 6096 . . . . . . . . 9
1110oveq1d 6096 . . . . . . . 8
1211mpt2eq3dva 6138 . . . . . . 7
13 eqid 2436 . . . . . . 7
14 seqomeq12 6711 . . . . . . 7 seq𝜔 seq𝜔
1512, 13, 14sylancl 644 . . . . . 6 seq𝜔 seq𝜔
1615fveq1d 5730 . . . . 5 seq𝜔 seq𝜔
1716csbeq2dv 3276 . . . 4 OrdIso seq𝜔 OrdIso seq𝜔
187, 17mpteq12dv 4287 . . 3 OrdIso seq𝜔 OrdIso seq𝜔
19 df-cnf 7617 . . 3 CNF OrdIso seq𝜔
20 ovex 6106 . . . . . 6
2120rabex 4354 . . . . 5
226, 21eqeltri 2506 . . . 4
2322mptex 5966 . . 3 OrdIso seq𝜔
2418, 19, 23ovmpt2a 6204 . 2 CNF OrdIso seq𝜔
251, 2, 24syl2anc 643 1 CNF OrdIso seq𝜔
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  crab 2709  cvv 2956  csb 3251   cdif 3317  c0 3628   cmpt 4266   cep 4492  con0 4581  ccnv 4877   cdm 4878  cima 4881  cfv 5454  (class class class)co 6081   cmpt2 6083  seq𝜔cseqom 6704  c1o 6717   coa 6721   comu 6722   coe 6723   cmap 7018  cfn 7109  OrdIsocoi 7478   CNF ccnf 7616 This theorem is referenced by:  cantnfdm  7619  cantnfval  7623  cantnff  7629 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-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403 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-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-seqom 6705  df-cnf 7617
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