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Theorem cnfcom3c 8603

Description: Wrap the construction of cnfcom3 8601 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.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 CNF CNF
2		eqid 2622	. 2 CNF CNF
3		eqid 2622	. 2 OrdIso CNF supp OrdIso CNF supp
4		eqid 2622	. 2 seq_𝜔 OrdIso CNF supp CNF OrdIso CNF supp seq_𝜔 OrdIso CNF supp CNF OrdIso CNF supp
5		eqid 2622	. 2 seq_𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp seq_𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp
6		eqid 2622	. 2 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp
7		eqid 2622	. 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 2622	. 2 OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp OrdIso CNF supp
9		eqid 2622	. 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 2622	. 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 2622	. 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 seq_𝜔 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 seq_𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp
12		eqid 2622	. 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 seq_𝜔 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 seq_𝜔 OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp CNF OrdIso CNF supp OrdIso CNF supp
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cnfcom3clem 8602	1 $\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wex 1704

wcel 1990

wral 2912

wrex 2913

cvv 3200 $\$ cdif 3571

cun 3572

wss 3574

c0 3915

cuni 4436

cmpt 4729

cep 5028

ccnv 5113

cdm 5114

ccom 5118

con0 5723

wf1o 5887

cfv 5888 (class class class)co 6650

cmpt2 6652

com 7065 supp csupp 7295 seq_𝜔cseqom 7542

c1o 7553

coa 7557

comu 7558

coe 7559 OrdIsocoi 8414 CNF ccnf 8558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seqom 7543 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-oexp 7566 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-cnf 8559

This theorem is referenced by: infxpenc2 8845

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