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Theorem cantnffval 8081
 Description: The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
Hypotheses
Ref Expression
cantnffval.s finSupp
cantnffval.a
cantnffval.b
Assertion
Ref Expression
cantnffval CNF OrdIso supp 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.a . 2
2 cantnffval.b . 2
3 oveq12 6294 . . . . . 6
4 rabeq 3107 . . . . . 6 finSupp finSupp
53, 4syl 16 . . . . 5 finSupp finSupp
6 cantnffval.s . . . . 5 finSupp
75, 6syl6eqr 2526 . . . 4 finSupp
8 simp1l 1020 . . . . . . . . . . 11
98oveq1d 6300 . . . . . . . . . 10
109oveq1d 6300 . . . . . . . . 9
1110oveq1d 6300 . . . . . . . 8
1211mpt2eq3dva 6346 . . . . . . 7
13 eqid 2467 . . . . . . 7
14 seqomeq12 7120 . . . . . . 7 seq𝜔 seq𝜔
1512, 13, 14sylancl 662 . . . . . 6 seq𝜔 seq𝜔
1615fveq1d 5868 . . . . 5 seq𝜔 seq𝜔
1716csbeq2dv 3835 . . . 4 OrdIso supp seq𝜔 OrdIso supp seq𝜔
187, 17mpteq12dv 4525 . . 3 finSupp OrdIso supp seq𝜔 OrdIso supp seq𝜔
19 df-cnf 8080 . . 3 CNF finSupp OrdIso supp seq𝜔
20 ovex 6310 . . . . . 6
2120rabex 4598 . . . . 5 finSupp
226, 21eqeltri 2551 . . . 4
2322mptex 6132 . . 3 OrdIso supp seq𝜔
2418, 19, 23ovmpt2a 6418 . 2 CNF OrdIso supp seq𝜔
251, 2, 24syl2anc 661 1 CNF OrdIso supp seq𝜔
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1379   wcel 1767  crab 2818  cvv 3113  csb 3435  c0 3785   class class class wbr 4447   cmpt 4505   cep 4789  con0 4878   cdm 4999  cfv 5588  (class class class)co 6285   cmpt2 6287   supp csupp 6902  seq𝜔cseqom 7113   coa 7128   comu 7129   coe 7130   cmap 7421   finSupp cfsupp 7830  OrdIsocoi 7935   CNF ccnf 8079 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-recs 7043  df-rdg 7077  df-seqom 7114  df-cnf 8080 This theorem is referenced by:  cantnfdm  8082  cantnffvalOLD  8083  cantnfval  8088  cantnff  8094
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