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Theorem frec0g 6543

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 4937	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2802	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4509	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15
6	5	nesymi 2446	. . . . . . . . . . . . . 14
7	1	eqeq1i 2237	. . . . . . . . . . . . . 14
8	6, 7	mtbir 675	. . . . . . . . . . . . 13
9	8	intnanr 935	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2622	. . . . . . . . . 10 $/\$
12	11	biorfi 751	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 733	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 206	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2345	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2350	. . . . . . 7
17	15, 16	eqtr3i 2252	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2811	. . . . . 6
19	17, 18	eqeltrid 2316	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4211	. . . . . . 7
21		dmeq 4923	. . . . . . . . . . . . 13
22	21	eqeq1d 2238	. . . . . . . . . . . 12
23		fveq1 5626	. . . . . . . . . . . . . 14
24	23	fveq2d 5631	. . . . . . . . . . . . 13
25	24	eleq2d 2299	. . . . . . . . . . . 12
26	22, 25	anbi12d 473	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2531	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2238	. . . . . . . . . . 11
29	28	anbi1d 465	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 798	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2347	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2229	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5710	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 424	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	eqtrdi 2278	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2306	. . 3 $/\$ $\/$ $/\$
38		df-frec 6537	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5628	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4686	. . . . . 6
41		fvres 5651	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 5	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2250	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2229	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 6469	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	eqtrid 2274	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2262	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 713

wceq 1395

wcel 2200

cab 2215

wne 2400

wrex 2509

cvv 2799

c0 3491

cmpt 4145

csuc 4456

com 4682

cdm 4719

cres 4721

cfv 5318 recscrecs 6450 freccfrec 6536

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-iord 4457 df-on 4459 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-res 4731 df-iota 5278 df-fun 5320 df-fn 5321 df-fv 5326 df-recs 6451 df-frec 6537

This theorem is referenced by: frecrdg 6554 frec2uz0d 10621 frec2uzrdg 10631 frecuzrdg0 10635 frecuzrdgg 10638 frecuzrdg0t 10644 seq3val 10682 seqvalcd 10683

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