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Theorem frec0g 6397

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 4841	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2740	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4418	. . . . . . . . . . . . . . . 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 4130	. . . . . . 7
21		dmeq 4827	. . . . . . . . . . . . 13
22	21	eqeq1d 2186	. . . . . . . . . . . 12
23		fveq1 5514	. . . . . . . . . . . . . 14
24	23	fveq2d 5519	. . . . . . . . . . . . 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 5592	. . . . . . 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 6391	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5516	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4593	. . . . . 6
41		fvres 5539	. . . . . 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 6323	. . . 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 4064

csuc 4365

com 4589

cdm 4626

cres 4628

cfv 5216 recscrecs 6304 freccfrec 6390

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 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536

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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-recs 6305 df-frec 6391

This theorem is referenced by: frecrdg 6408 frec2uz0d 10398 frec2uzrdg 10408 frecuzrdg0 10412 frecuzrdgg 10415 frecuzrdg0t 10421 seq3val 10457 seqvalcd 10458

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