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Theorem frec0g 6398

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 4842	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2741	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4419	. . . . . . . . . . . . . . . 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 2749	. . . . . 6
19	17, 18	eqeltrid 2264	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4131	. . . . . . 7
21		dmeq 4828	. . . . . . . . . . . . 13
22	21	eqeq1d 2186	. . . . . . . . . . . 12
23		fveq1 5515	. . . . . . . . . . . . . 14
24	23	fveq2d 5520	. . . . . . . . . . . . 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 5593	. . . . . . 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 6392	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5517	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4594	. . . . . 6
41		fvres 5540	. . . . . 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 6324	. . . 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 2738

c0 3423

cmpt 4065

csuc 4366

com 4590

cdm 4627

cres 4629

cfv 5217 recscrecs 6305 freccfrec 6391

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 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537

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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-iord 4367 df-on 4369 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-res 4639 df-iota 5179 df-fun 5219 df-fn 5220 df-fv 5225 df-recs 6306 df-frec 6392

This theorem is referenced by: frecrdg 6409 frec2uz0d 10399 frec2uzrdg 10409 frecuzrdg0 10413 frecuzrdgg 10416 frecuzrdg0t 10422 seq3val 10458 seqvalcd 10459

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