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Theorem frec0g 6658

Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)

**Assertion**
Ref	Expression
frec0g	frec

Proof of Theorem frec0g Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dm0 4990	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2824	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4558	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15
6	5	nesymi 2466	. . . . . . . . . . . . . 14
7	1	eqeq1i 2246	. . . . . . . . . . . . . 14
8	6, 7	mtbir 682	. . . . . . . . . . . . 13
9	8	intnanr 942	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2642	. . . . . . . . . 10 $/\$
12	11	biorfi 758	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 740	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 206	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2354	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2361	. . . . . . 7
17	15, 16	eqtr3i 2261	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2833	. . . . . 6
19	17, 18	eqeltrid 2325	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4255	. . . . . . 7
21		dmeq 4976	. . . . . . . . . . . . 13
22	21	eqeq1d 2247	. . . . . . . . . . . 12
23		fveq1 5689	. . . . . . . . . . . . . 14
24	23	fveq2d 5694	. . . . . . . . . . . . 13
25	24	eleq2d 2308	. . . . . . . . . . . 12
26	22, 25	anbi12d 477	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2551	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2247	. . . . . . . . . . 11
29	28	anbi1d 469	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 805	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2358	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2238	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5775	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 428	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	eqtrdi 2287	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2315	. . 3 $/\$ $\/$ $/\$
38		df-frec 6652	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5691	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4736	. . . . . 6
41		fvres 5714	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 5	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2259	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2238	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 6584	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	eqtrid 2283	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2271	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 720

wceq 1402

wcel 2209

cab 2224

wne 2420

wrex 2529

cvv 2821

c0 3520

cmpt 4187

csuc 4505

com 4732

cdm 4769

cres 4771

cfv 5372 recscrecs 6565 freccfrec 6651

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-iord 4506 df-on 4508 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-res 4781 df-iota 5332 df-fun 5374 df-fn 5375 df-fv 5380 df-recs 6566 df-frec 6652

This theorem is referenced by: frecrdg 6669 frec2uz0d 10814 frec2uzrdg 10824 frecuzrdg0 10828 frecuzrdgg 10831 frecuzrdg0t 10837 seq3val 10875 seqvalcd 10876

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