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Theorem frec0g 6287

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 4748	. . . . . . . . . 10
2	1	biantrur 301	. . . . . . . . 9 $/\$
3		vex 2684	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4335	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15
6	5	nesymi 2352	. . . . . . . . . . . . . 14
7	1	eqeq1i 2145	. . . . . . . . . . . . . 14
8	6, 7	mtbir 660	. . . . . . . . . . . . 13
9	8	intnanr 915	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2522	. . . . . . . . . 10 $/\$
12	11	biorfi 735	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 717	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 205	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2253	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2258	. . . . . . 7
17	15, 16	eqtr3i 2160	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2692	. . . . . 6
19	17, 18	eqeltrid 2224	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4050	. . . . . . 7
21		dmeq 4734	. . . . . . . . . . . . 13
22	21	eqeq1d 2146	. . . . . . . . . . . 12
23		fveq1 5413	. . . . . . . . . . . . . 14
24	23	fveq2d 5418	. . . . . . . . . . . . 13
25	24	eleq2d 2207	. . . . . . . . . . . 12
26	22, 25	anbi12d 464	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2436	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2146	. . . . . . . . . . 11
29	28	anbi1d 460	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 782	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2255	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2137	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5490	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 420	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	syl6eq 2186	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2214	. . 3 $/\$ $\/$ $/\$
38		df-frec 6281	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5415	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4503	. . . . . 6
41		fvres 5438	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 5	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2158	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2137	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 6213	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	syl5eq 2182	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2170	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 697

wceq 1331

wcel 1480

cab 2123

wne 2306

wrex 2415

cvv 2681

c0 3358

cmpt 3984

csuc 4282

com 4499

cdm 4534

cres 4536

cfv 5118 recscrecs 6194 freccfrec 6280

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-res 4546 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-recs 6195 df-frec 6281

This theorem is referenced by: frecrdg 6298 frec2uz0d 10165 frec2uzrdg 10175 frecuzrdg0 10179 frecuzrdgg 10182 frecuzrdg0t 10188 seq3val 10224 seqvalcd 10225

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