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Theorem frec0g 6464

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 4881	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2766	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4454	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15
6	5	nesymi 2413	. . . . . . . . . . . . . 14
7	1	eqeq1i 2204	. . . . . . . . . . . . . 14
8	6, 7	mtbir 672	. . . . . . . . . . . . 13
9	8	intnanr 931	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2589	. . . . . . . . . 10 $/\$
12	11	biorfi 747	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 729	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 206	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2312	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2317	. . . . . . 7
17	15, 16	eqtr3i 2219	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2774	. . . . . 6
19	17, 18	eqeltrid 2283	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4161	. . . . . . 7
21		dmeq 4867	. . . . . . . . . . . . 13
22	21	eqeq1d 2205	. . . . . . . . . . . 12
23		fveq1 5560	. . . . . . . . . . . . . 14
24	23	fveq2d 5565	. . . . . . . . . . . . 13
25	24	eleq2d 2266	. . . . . . . . . . . 12
26	22, 25	anbi12d 473	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2498	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2205	. . . . . . . . . . 11
29	28	anbi1d 465	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 794	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2314	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2196	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5640	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 424	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	eqtrdi 2245	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2273	. . 3 $/\$ $\/$ $/\$
38		df-frec 6458	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5562	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4631	. . . . . 6
41		fvres 5585	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 5	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2217	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2196	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 6390	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	eqtrid 2241	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2229	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 709

wceq 1364

wcel 2167

cab 2182

wne 2367

wrex 2476

cvv 2763

c0 3451

cmpt 4095

csuc 4401

com 4627

cdm 4664

cres 4666

cfv 5259 recscrecs 6371 freccfrec 6457

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-iord 4402 df-on 4404 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-res 4676 df-iota 5220 df-fun 5261 df-fn 5262 df-fv 5267 df-recs 6372 df-frec 6458

This theorem is referenced by: frecrdg 6475 frec2uz0d 10508 frec2uzrdg 10518 frecuzrdg0 10522 frecuzrdgg 10525 frecuzrdg0t 10531 seq3val 10569 seqvalcd 10570

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