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Theorem frec0g 6401

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 4843	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2742	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4420	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15
6	5	nesymi 2393	. . . . . . . . . . . . . 14
7	1	eqeq1i 2185	. . . . . . . . . . . . . 14
8	6, 7	mtbir 671	. . . . . . . . . . . . 13
9	8	intnanr 930	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2569	. . . . . . . . . 10 $/\$
12	11	biorfi 746	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 728	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 206	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2293	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2298	. . . . . . 7
17	15, 16	eqtr3i 2200	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2750	. . . . . 6
19	17, 18	eqeltrid 2264	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4132	. . . . . . 7
21		dmeq 4829	. . . . . . . . . . . . 13
22	21	eqeq1d 2186	. . . . . . . . . . . 12
23		fveq1 5516	. . . . . . . . . . . . . 14
24	23	fveq2d 5521	. . . . . . . . . . . . 13
25	24	eleq2d 2247	. . . . . . . . . . . 12
26	22, 25	anbi12d 473	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2478	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2186	. . . . . . . . . . 11
29	28	anbi1d 465	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 793	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2295	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2177	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5595	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 424	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	eqtrdi 2226	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2254	. . 3 $/\$ $\/$ $/\$
38		df-frec 6395	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5518	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4595	. . . . . 6
41		fvres 5541	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 5	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2198	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2177	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 6327	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	eqtrid 2222	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2210	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 708

wceq 1353

wcel 2148

cab 2163

wne 2347

wrex 2456

cvv 2739

c0 3424

cmpt 4066

csuc 4367

com 4591

cdm 4628

cres 4630

cfv 5218 recscrecs 6308 freccfrec 6394

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-res 4640 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-recs 6309 df-frec 6395

This theorem is referenced by: frecrdg 6412 frec2uz0d 10402 frec2uzrdg 10412 frecuzrdg0 10416 frecuzrdgg 10419 frecuzrdg0t 10425 seq3val 10461 seqvalcd 10462

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