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Theorem cnfcom3c 8210
 Description: Wrap the construction of cnfcom3 8208 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.)
Assertion
Ref Expression
cnfcom3c
Distinct variable group:   ,,,

Proof of Theorem cnfcom3c
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . 2 CNF CNF
2 eqid 2429 . 2 CNF CNF
3 eqid 2429 . 2 OrdIso CNF supp OrdIso CNF supp
4 eqid 2429 . 2 seq𝜔 OrdIso CNF supp CNF OrdIso CNF supp seq𝜔 OrdIso CNF supp CNF OrdIso CNF supp
5 eqid 2429 . 2 seq𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp seq𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp
6 eqid 2429 . 2 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp
7 eqid 2429 . 2 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp
8 eqid 2429 . 2 OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp
9 eqid 2429 . 2 CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp
10 eqid 2429 . 2 CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp
11 eqid 2429 . 2 CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp seq𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp seq𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp
12 eqid 2429 . 2 CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp seq𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp seq𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cnfcom3clem 8209 1
 Colors of variables: wff setvar class Syntax hints:   wi 4  wex 1659   wcel 1870  wral 2782  wrex 2783  cvv 3087   cdif 3439   cun 3440   wss 3442  c0 3767  cuni 4222   cmpt 4484   cep 4763  ccnv 4853   cdm 4854   ccom 4858  con0 5442  wf1o 5600  cfv 5601  (class class class)co 6305   cmpt2 6307  com 6706   supp csupp 6925  seq𝜔cseqom 7172  c1o 7183   coa 7187   comu 7188   coe 7189  OrdIsocoi 8024   CNF ccnf 8165 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-seqom 7173  df-1o 7190  df-2o 7191  df-oadd 7194  df-omul 7195  df-oexp 7196  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-oi 8025  df-cnf 8166 This theorem is referenced by:  infxpenc2  8451
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