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Theorem frec0g 6455

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 4880	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2766	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4453	. . . . . . . . . . . . . . . 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 4160	. . . . . . 7
21		dmeq 4866	. . . . . . . . . . . . 13
22	21	eqeq1d 2205	. . . . . . . . . . . 12
23		fveq1 5557	. . . . . . . . . . . . . 14
24	23	fveq2d 5562	. . . . . . . . . . . . 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 5637	. . . . . . 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 6449	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5559	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4630	. . . . . 6
41		fvres 5582	. . . . . 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 6381	. . . 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 3450

cmpt 4094

csuc 4400

com 4626

cdm 4663

cres 4665

cfv 5258 recscrecs 6362 freccfrec 6448

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 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573

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 3451 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-res 4675 df-iota 5219 df-fun 5260 df-fn 5261 df-fv 5266 df-recs 6363 df-frec 6449

This theorem is referenced by: frecrdg 6466 frec2uz0d 10491 frec2uzrdg 10501 frecuzrdg0 10505 frecuzrdgg 10508 frecuzrdg0t 10514 seq3val 10552 seqvalcd 10553

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