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Theorem frec0g 6483

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 4892	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2775	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4465	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15
6	5	nesymi 2422	. . . . . . . . . . . . . 14
7	1	eqeq1i 2213	. . . . . . . . . . . . . 14
8	6, 7	mtbir 673	. . . . . . . . . . . . 13
9	8	intnanr 932	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2598	. . . . . . . . . 10 $/\$
12	11	biorfi 748	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 730	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 206	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2321	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2326	. . . . . . 7
17	15, 16	eqtr3i 2228	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2783	. . . . . 6
19	17, 18	eqeltrid 2292	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4171	. . . . . . 7
21		dmeq 4878	. . . . . . . . . . . . 13
22	21	eqeq1d 2214	. . . . . . . . . . . 12
23		fveq1 5575	. . . . . . . . . . . . . 14
24	23	fveq2d 5580	. . . . . . . . . . . . 13
25	24	eleq2d 2275	. . . . . . . . . . . 12
26	22, 25	anbi12d 473	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2507	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2214	. . . . . . . . . . 11
29	28	anbi1d 465	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 795	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2323	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2205	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5655	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 424	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	eqtrdi 2254	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2282	. . 3 $/\$ $\/$ $/\$
38		df-frec 6477	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5577	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4642	. . . . . 6
41		fvres 5600	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 5	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2226	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2205	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 6409	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	eqtrid 2250	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2238	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 710

wceq 1373

wcel 2176

cab 2191

wne 2376

wrex 2485

cvv 2772

c0 3460

cmpt 4105

csuc 4412

com 4638

cdm 4675

cres 4677

cfv 5271 recscrecs 6390 freccfrec 6476

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-iord 4413 df-on 4415 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-res 4687 df-iota 5232 df-fun 5273 df-fn 5274 df-fv 5279 df-recs 6391 df-frec 6477

This theorem is referenced by: frecrdg 6494 frec2uz0d 10544 frec2uzrdg 10554 frecuzrdg0 10558 frecuzrdgg 10561 frecuzrdg0t 10567 seq3val 10605 seqvalcd 10606

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