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Theorem cantnfdm 7611
 Description: The domain of the Cantor normal form function (in later lemmas we will use CNF to abbreviate "the set of finitely supported functions from to "). (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnffval.1
cantnffval.2
cantnffval.3
Assertion
Ref Expression
cantnfdm CNF
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem cantnfdm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnffval.1 . . . 4
2 cantnffval.2 . . . 4
3 cantnffval.3 . . . 4
41, 2, 3cantnffval 7610 . . 3 CNF OrdIso seq𝜔
54dmeqd 5064 . 2 CNF OrdIso seq𝜔
6 vex 2951 . . . . . . 7
76cnvex 5398 . . . . . 6
8 imaexg 5209 . . . . . 6
9 eqid 2435 . . . . . . 7 OrdIso OrdIso
109oiexg 7496 . . . . . 6 OrdIso
117, 8, 10mp2b 10 . . . . 5 OrdIso
12 fvex 5734 . . . . 5 seq𝜔
1311, 12csbex 3254 . . . 4 OrdIso seq𝜔
1413rgenw 2765 . . 3 OrdIso seq𝜔
15 dmmptg 5359 . . 3 OrdIso seq𝜔 OrdIso seq𝜔
1614, 15ax-mp 8 . 2 OrdIso seq𝜔
175, 16syl6eq 2483 1 CNF
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  wral 2697  crab 2701  cvv 2948  csb 3243   cdif 3309  c0 3620   cmpt 4258   cep 4484  con0 4573  ccnv 4869   cdm 4870  cima 4873  cfv 5446  (class class class)co 6073   cmpt2 6075  seq𝜔cseqom 6696  c1o 6709   coa 6713   comu 6714   coe 6715   cmap 7010  cfn 7101  OrdIsocoi 7470   CNF ccnf 7608 This theorem is referenced by:  cantnfs  7613  cantnfval  7615  cantnff  7621  oemapso  7630  wemapwe  7646  oef1o  7647 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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-seqom 6697  df-oi 7471  df-cnf 7609
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