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Theorem cnfcom3cOLD 8189
 Description: Wrap the construction of cnfcom3OLD 8187 into an existence quantifier. For any , there is a bijection from to some power of . Furthermore, this bijection is canonical , which means that we can find a single function which will give such bijections for every less than some arbitrarily large bound . (Contributed by Mario Carneiro, 30-May-2015.) Obsolete version of cnfcom3c 8181 as of 4-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
cnfcom3cOLD
Distinct variable group:   ,,,

Proof of Theorem cnfcom3cOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . 2 CNF CNF
2 eqid 2402 . 2 CNF CNF
3 eqid 2402 . 2 OrdIso CNF OrdIso CNF
4 eqid 2402 . 2 seq𝜔 OrdIso CNF CNF OrdIso CNF seq𝜔 OrdIso CNF CNF OrdIso CNF
5 eqid 2402 . 2 seq𝜔 OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF seq𝜔 OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF
6 eqid 2402 . 2 OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF
7 eqid 2402 . 2 OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF
8 eqid 2402 . 2 OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF
9 eqid 2402 . 2 CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF
10 eqid 2402 . 2 CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF
11 eqid 2402 . 2 CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF seq𝜔 OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF seq𝜔 OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF
12 eqid 2402 . 2 CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF seq𝜔 OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF OrdIso CNF seq𝜔 OrdIso CNF CNF OrdIso CNF OrdIso CNF CNF OrdIso CNF OrdIso CNF
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cnfcom3clemOLD 8188 1
 Colors of variables: wff setvar class Syntax hints:   wi 4  wex 1633   wcel 1842  wral 2753  wrex 2754  cvv 3058   cdif 3410   cun 3411   wss 3413  c0 3737  cuni 4190   cmpt 4452   cep 4731  ccnv 4821   cdm 4822  cima 4825   ccom 4826  con0 5409  wf1o 5567  cfv 5568  (class class class)co 6277   cmpt2 6279  com 6682  seq𝜔cseqom 7148  c1o 7159   coa 7163   comu 7164   coe 7165  OrdIsocoi 7967   CNF ccnf 8109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-seqom 7149  df-1o 7166  df-2o 7167  df-oadd 7170  df-omul 7171  df-oexp 7172  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-oi 7968  df-cnf 8110 This theorem is referenced by:  infxpenc2OLD  8434
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