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Theorem frec0g 6641

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 4975	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2818	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4544	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15
6	5	nesymi 2460	. . . . . . . . . . . . . 14
7	1	eqeq1i 2242	. . . . . . . . . . . . . 14
8	6, 7	mtbir 678	. . . . . . . . . . . . 13
9	8	intnanr 938	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2636	. . . . . . . . . 10 $/\$
12	11	biorfi 754	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 736	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 206	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2350	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2357	. . . . . . 7
17	15, 16	eqtr3i 2257	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2827	. . . . . 6
19	17, 18	eqeltrid 2321	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4242	. . . . . . 7
21		dmeq 4961	. . . . . . . . . . . . 13
22	21	eqeq1d 2243	. . . . . . . . . . . 12
23		fveq1 5674	. . . . . . . . . . . . . 14
24	23	fveq2d 5679	. . . . . . . . . . . . 13
25	24	eleq2d 2304	. . . . . . . . . . . 12
26	22, 25	anbi12d 473	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2545	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2243	. . . . . . . . . . 11
29	28	anbi1d 465	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 801	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2354	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2234	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5758	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 424	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	eqtrdi 2283	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2311	. . 3 $/\$ $\/$ $/\$
38		df-frec 6635	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5676	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4721	. . . . . 6
41		fvres 5699	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 5	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2255	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2234	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 6567	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	eqtrid 2279	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2267	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 716

wceq 1398

wcel 2205

cab 2220

wne 2414

wrex 2523

cvv 2815

c0 3512

cmpt 4176

csuc 4491

com 4717

cdm 4754

cres 4756

cfv 5357 recscrecs 6548 freccfrec 6634

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-res 4766 df-iota 5317 df-fun 5359 df-fn 5360 df-fv 5365 df-recs 6549 df-frec 6635

This theorem is referenced by: frecrdg 6652 frec2uz0d 10785 frec2uzrdg 10795 frecuzrdg0 10799 frecuzrdgg 10802 frecuzrdg0t 10808 seq3val 10846 seqvalcd 10847

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