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Theorem frec0g 6392

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 4837	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2740	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4415	. . . . . . . . . . . . . . . 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 2748	. . . . . 6
19	17, 18	eqeltrid 2264	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4127	. . . . . . 7
21		dmeq 4823	. . . . . . . . . . . . 13
22	21	eqeq1d 2186	. . . . . . . . . . . 12
23		fveq1 5510	. . . . . . . . . . . . . 14
24	23	fveq2d 5515	. . . . . . . . . . . . 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 5588	. . . . . . 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 6386	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5512	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4590	. . . . . 6
41		fvres 5535	. . . . . 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 6318	. . . 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 2737

c0 3422

cmpt 4061

csuc 4362

com 4586

cdm 4623

cres 4625

cfv 5212 recscrecs 6299 freccfrec 6385

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 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533

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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-res 4635 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-recs 6300 df-frec 6386

This theorem is referenced by: frecrdg 6403 frec2uz0d 10385 frec2uzrdg 10395 frecuzrdg0 10399 frecuzrdgg 10402 frecuzrdg0t 10408 seq3val 10444 seqvalcd 10445

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